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# Random walks on simplicial complexes

Thomas Bonis, Laurent Decreusefond, Viet Chi Tran, Iris Zhihan Zhang

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Thomas Bonis, Laurent Decreusefond, Viet Chi Tran, Iris Zhihan Zhang. Random walks on simplicial complexes. 2024. hal-04544608v2

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Preprint submitted on 7 May 2024

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# Random walks on simplicial complexes

Thomas Bonis,\* Laurent Decreusefond,† Viet Chi Tran,‡ Zhihan Iris Zhang §

May 7, 2024

## Abstract

The notion of Laplacian of a graph can be generalized to simplicial complexes and hypergraphs, and contains information on the topology of these structures. Even for a graph, the consideration of associated simplicial complexes is interesting to understand its shape. Whereas the Laplacian of a graph has a simple probabilistic interpretation as the generator of a continuous time Markov chain on the graph, things are not so direct when considering simplicial complexes. We define here new Markov chains on simplicial complexes. For a given order $k$, the state space is the set of $k$-cycles that are chains of $k$-simplexes with null boundary. This new framework is a natural generalization of the canonical Markov chains on graphs. We show that the generator of our Markov chain is the upper Laplacian defined in the context of algebraic topology for discrete structure. We establish several key properties of this new process: in particular, when the number of vertices is finite, the Markov chain is positive recurrent. We study the diffusive limits when the simplicial complexes under scrutiny are a sequence of ever refining triangulations of the flat torus. Using the analogy between singular and Hodge homologies, we express this limit as valued in the set of currents. The proof of tightness and the identification of the limiting martingale problem make use of the flat norm and carefully controls of the error terms in the convergence of the generator. Uniqueness of the solution to the martingale problem is left open. An application to hole detection is carried.

**keywords:** stochastic geometry; random walk; simplicial complex; algebraic topology; Laplacian of a simplicial complex; homology; limit theorem; Hodge diffusion.

**Subclass:** 60D05.

**Acknowledgements:** L.D. would like to thank A. Vergne, M. Glisse, O. Devillers and P. Chassaing for fruitful discussions. V.C.T. thanks P. Melotti for inspiring simulations. This work benefited from the GdR GeoSto 3477. T.B., L.D. and Z.Z. acknowledge support from ANR ASPAG (grant ANR-17-CE40-0017 of the French National Research Agency). V.C.T. is supported by Labex Bézout (ANR-10-LABX-58), ANR Econet (ANR-18-CE02-0010) and by the European Union (ERC-AdG SINGER-101054787).

## 1 Introduction

Exploring and understanding complex structures such as graphs is a difficult and rich problem that has motivated an abundant literature in the last years. Random walks are one of the many tools used to study the connectivity of a graph. For instance, the PageRank algorithm [42] uses invariant measures of random walks to highlight central nodes of graphs. Random walks on graphs are also used to provide distances between nodes by considering the expected time required for the walk to travel between nodes. This metric is called the commute distance and has been used in a large number of applications such as graph embedding [57], semi-supervised learning [61], clustering [58] and many more, see for instance the introduction of [54] for a more complete list. Finally, diffusions of random walks on graphs is at the center of the diffusion maps graph embedding approach [11].

<footnote>\*LAMA, Univ Gustave Eiffel, Univ Paris Est Creteil, CNRS, F-77454 Marne-la-Vallée, France.
thomas.bonis@univ-eiffel.fr</footnote>

<footnote>†LTCI, Telecom Paris, I.P. Paris, France. laurent.decreasefond@telecom-paris.fr</footnote>

<footnote>‡LAMA, Univ Gustave Eiffel, Univ Paris Est Creteil, CNRS, F-77454 Marne-la-Vallée, France; IRL 3457, CRM-CNRS, Universit de Montral, Canada. chi.tran@univ-eiffel.fr</footnote>

<footnote>§LTCI, Telecom Paris, I.P. Paris, France. zhihan.zhang@telecom-paris.fr</footnote>
# Random walks on simplicial complexes

Thomas Bonis,\* Laurent Decreusefond,† Viet Chi Tran,‡ Zhihan Iris Zhang §

May 7, 2024

## Abstract

The notion of Laplacian of a graph can be generalized to simplicial complexes and hypergraphs, and contains information on the topology of these structures. Even for a graph, the consideration of associated simplicial complexes is interesting to understand its shape. Whereas the Laplacian of a graph has a simple probabilistic interpretation as the generator of a continuous time Markov chain on the graph, things are not so direct when considering simplicial complexes. We define here new Markov chains on simplicial complexes. For a given order $k$, the state space is the set of $k$-cycles that are chains of $k$-simplexes with null boundary. This new framework is a natural generalization of the canonical Markov chains on graphs. We show that the generator of our Markov chain is the upper Laplacian defined in the context of algebraic topology for discrete structure. We establish several key properties of this new process: in particular, when the number of vertices is finite, the Markov chain is positive recurrent. We study the diffusive limits when the simplicial complexes under scrutiny are a sequence of ever refining triangulations of the flat torus. Using the analogy between singular and Hodge homologies, we express this limit as valued in the set of currents. The proof of tightness and the identification of the limiting martingale problem make use of the flat norm and carefully controls of the error terms in the convergence of the generator. Uniqueness of the solution to the martingale problem is left open. An application to hole detection is carried.

**keywords:** stochastic geometry; random walk; simplicial complex; algebraic topology; Laplacian of a simplicial complex; homology; limit theorem; Hodge diffusion.

**Subclass:** 60D05.

**Acknowledgements:** L.D. would like to thanks A. Vergne, M. Glisse, O. Devillers and P. Chassaing for fruitful discussions. V.C.T. thanks P. Melotti for inspiring simulations. This work benefited from the GdR GeoSto 3477. T.B., L.D. and Z.Z. acknowledge support from ANR ASPAG (grant ANR-17-CE40-0017 of the French National Research Agency). V.C.T. is supported by Labex Bézout (ANR-10-LABX-58), ANR Econet (ANR-18-CE02-0010) and by the European Union (ERC-AdG SINGER-101054787).

## 1 Introduction

Exploring and understanding complex structures such as graphs is a difficult and rich problem that has motivated an abundant literature in the last years. Random walks are one of the many tools used to study the connectivity of a graph. For instance, the PageRank algorithm [42] uses invariant measures of random walks to highlight central nodes of graphs. Random walks on graphs are also used to provide distances between nodes by considering the expected time required for the walk to travel between nodes. This metric is called the commute distance and has been used in a large number of applications such as graph embedding [57], semi-supervised learning [61], clustering [58] and many more, see for instance the introduction of [54] for a more complete list. Finally, diffusions of random walks on graphs is at the center of the diffusion maps graph embedding approach [11].

<footnote>\*LAMA, Univ Gustave Eiffel, Univ Paris Est Creteil, CNRS, F-77454 Marne-la-Vallée, France.
thomas.bonis@univ-eiffel.fr</footnote>

<footnote>†LTCI, Telecom Paris, I.P. Paris, France. laurent.decreasefond@telecom-paris.fr</footnote>

<footnote>‡LAMA, Univ Gustave Eiffel, Univ Paris Est Creteil, CNRS, F-77454 Marne-la-Vallée, France; IRL 3457, CRM-CNRS, Universit de Montral, Canada. chi.tran@univ-eiffel.fr</footnote>

<footnote>§LTCI, Telecom Paris, I.P. Paris, France. zhihan.zhang@telecom-paris.fr</footnote>
The reason random walks on graphs are so tightly linked to the connectivity structure of the graph can be found in their generators. Consider a finite non-oriented graph $G = (V, E)$ consisting of the (finite) sets of vertices $V$ and edges $E$ determining the pairs of vertices that are connected. The adjacency matrix $A$ of $G$ is defined as the matrix whose entry at line $u$ and column $v$ is 1 if and only if $(u, v)$ is an edge of $G$, which we will denote by $u \sim v$. For any $u \in V$ and for any function $f$ from $V$ to $\mathbb{R}$, the generator of the random walk is

$$Af(u) = \sum_{v \sim u} (f(v) - f(u)) = -(D - A)f(u), \qquad (1)$$

where $D$ is the diagonal matrix containing the degrees of the vertices. Thus, the generator of the random walk is the opposite of the Laplacian of the graph, $L = D - A$. It is well-known that this Laplacian contains information regarding the connectivity structures of the graph. For instance, the dimension of its kernel is equal to the number of connected components of the graph while, more generally, small eigenvalues indicate almost disconnected components [53].

But connectivity is only a fraction of the topological information contained in complex structures. For instance, let us consider a circular-like graph presented in Figure 1(a). Such a topological structure is not well described through connectivity. More generally, graphs are not suited to deal with such structures since adding a single node to the graph creates another artifact circle which is not an actual feature of the data, see Figure 1(b). The correct structures one should use to deal with higher-order topology are simplicial complexes. The definition of simplicial complexes is recalled later, but a natural simplicial complex associated with a graph is its Rips-Vietoris simplicial complex [8] obtained by adding to the pair $(V, E)$ the set $S_2$ of all triangles whose edges belong to $E$, the set of all tetrahedrons $S_3$ whose triangles belong to $S_2$ etc. For instance, adding a triangle in the previous example, we recover a circular structure (see Figure 1(c)). This notion of circular structure is formalized by the concepts of homology classes and Betti number which we present in Section 2.1.

We are interested in generalizing the notion of random walks on graphs to random walk on simplicial complexes, hoping they can be used to derive new algorithms for topological data analysis by generalizing algorithms such as PageRank. Furthermore, providing a probabilistic interpretation of topological properties of simplicial complexes is important since there is a growing literature on the subject in recent years. While the relations between graphs and simplicial complexes are considered in [13], papers more focused on the simplicial complexes themselves include for example studies of the Betti numbers and volume-like computation for random clique complexes built over the Erdös-Rényi graphs [30, 31, 41] or Čech and Vietoris-Rips complexes built over stationary point processes [1, 12, 59], or computation of convex hulls of simplicial complexes [22].

It is already known that the graph Laplacian $D - A$ is a specific instance of the more general combinatorial Laplacian, introduced by Eckmann [15]. In a similar way that the graph Laplacian contains information regarding the connectivity of the graph, these combinatorial Laplacians describe the structure of the homology groups of the simplicial complex and are related to higher order Betti numbers. Since the generator of random walks on nodes of graphs is equal to the opposite graph Laplacian, it was proposed in [32, 43, 44, 46] to define random walks on simplicial complexes as random walks with generators equal to the opposite of the combinatorial Laplacians. As the combinatorial Laplacian is defined as a sum of two operators, called up-Laplacian and down-Laplacian, such an approach leads to defining two different random walks from which it is not clear how to generalize graph analysis algorithms. For instance, if a combinatorial Laplacian is associated to two different random walks and thus to two different invariant measures, which one should be preferred to obtain an equivalent of PageRank for simplicial complexes?

In this paper, we propose to define a random walk on a simplicial complex in a totally different way. More precisely, we consider a random walk on the space of cycles of the simplicial complex whose transitions are given by the very definition of homology groups which, incidentally, has the opposite of the combinatorial up-Laplacian as generator. In particular, similarly to how a random walk on a graph cannot leave a given connected component, which is a homology class of dimension 0, our random walk is bound to stay in the homology class of its initial state.

Let us detail our random walk in the case of a simplicial complex of order 2 (a general and precise
The reason random walks on graphs are so tightly linked to the connectivity structure of the graph can be found in their generators. Consider a finite non-oriented graph $G = (V, E)$ consisting of the (finite) sets of vertices $V$ and edges $E$ determining the pairs of vertices that are connected. The adjacency matrix $A$ of $G$ is defined as the matrix whose entry at line $u$ and column $v$ is 1 if and only if $(u, v)$ is an edge of $G$, which we will denote by $u \sim v$. For any $u \in V$ and for any function $f$ from $V$ to $\mathbb{R}$, the generator of the random walk is

$$Af(u) = \sum_{v \sim u} (f(v) - f(u)) = -(D - A)f(u), \qquad (1)$$

where $D$ is the diagonal matrix containing the degrees of the vertices. Thus, the generator of the random walk is the opposite of the Laplacian of the graph, $L = D - A$. It is well-known that this Laplacian contains information regarding the connectivity structures of the graph. For instance, the dimension of its kernel is equal to the number of connected components of the graph while, more generally, small eigenvalues indicate almost disconnected components [53].

But connectivity is only a fraction of the topological information contained in complex structures. For instance, let us consider a circular-like graph presented in Figure 1(a). Such a topological structure is not well described through connectivity. More generally, graphs are not suited to deal with such structures since adding a single node to the graph creates another artifact circle which is not an actual feature of the data, see Figure 1(b). The correct structures one should use to deal with higher-order topology are simplicial complexes. The definition of simplicial complexes is recalled later, but a natural simplicial complex associated with a graph is its Rips-Vietoris simplicial complex [8] obtained by adding to the pair $(V, E)$ the set $S_2$ of all triangles whose edges belong to $E$, the set of all tetrahedrons $S_3$ whose triangles belong to $S_2$ etc. For instance, adding a triangle in the previous example, we recover a circular structure (see Figure 1(c)). This notion of circular structure is formalized by the concepts of homology classes and Betti number which we present in Section 2.1.

We are interested in generalizing the notion of random walks on graphs to random walk on simplicial complexes, hoping they can be used to derive new algorithms for topological data analysis by generalizing algorithms such as PageRank. Furthermore, providing a probabilistic interpretation of topological properties of simplicial complexes is important since there is a growing literature on the subject in recent years. While the relations between graphs and simplicial complexes are considered in [13], papers more focused on the simplicial complexes themselves include for example studies of the Betti numbers and volume-like computation for random clique complexes built over the Erdös-Rényi graphs [30, 31, 41] or Čech and Vietoris-Rips complexes built over stationary point processes [1, 12, 59], or computation of convex hulls of simplicial complexes [22].

It is already known that the graph Laplacian $D - A$ is a specific instance of the more general combinatorial Laplacian, introduced by Eckmann [15]. In a similar way that the graph Laplacian contains information regarding the connectivity of the graph, these combinatorial Laplacians describe the structure of the homology groups of the simplicial complex and are related to higher order Betti numbers. Since the generator of random walks on nodes of graphs is equal to the opposite graph Laplacian, it was proposed in [32, 43, 44, 46] to define random walks on simplicial complexes as random walks with generators equal to the opposite of the combinatorial Laplacians. As the combinatorial Laplacian is defined as a sum of two operators, called up-Laplacian and down-Laplacian, such an approach leads to defining two different random walks from which it is not clear how to generalize graph analysis algorithms. For instance, if a combinatorial Laplacian is associated to two different random walks and thus to two different invariant measures, which one should be preferred to obtain an equivalent of PageRank for simplicial complexes?

In this paper, we propose to define a random walk on a simplicial complex in a totally different way. More precisely, we consider a random walk on the space of cycles of the simplicial complex whose transitions are given by the very definition of homology groups which, incidentally, has the opposite of the combinatorial up-Laplacian as generator. In particular, similarly to how a random walk on a graph cannot leave a given connected component, which is a homology class of dimension 0, our random walk is bound to stay in the homology class of its initial state.

Let us detail our random walk in the case of a simplicial complex of order 2 (a general and precise
description is given in the paper). Recall that a simplicial complex $\mathcal{C}$ of dimension 2 is a collection of vertices $V$, of edges $E$ and of triangles $T$ such that the edges of the triangles belong to $E$ and the vertices of the edges belong to $V$. Define the oriented edge from $u$ to $v$ as $[u, v]$. The cycles are defined as chains, i.e. elements of the vector space spanned by $E^+$ the set of edges with positive orientation:

$$ \mathcal{C}_1 = \left\{ \sum_{[u,v] \in E^+} \lambda_{[u,v]}[u,v], \text{ with } \forall [u,v] \in E^+, \lambda_{[u,v]} \in \mathbb{R} \right\}, $$

with the constraint that they have no border. Defining, the boundary map $\partial$ for edges by $\partial[u, v] = v - u$, and for triangles by $\partial[u, v, w] = [v, w] - [u, w] + [u, v]$, a chain $\sigma \in \mathcal{C}_1$ is a cycle if $\partial\sigma = 0$. Our random walk $(X_t)_{t \in \mathbb{R}_+}$ is a continuous time Markov chain whose state space consists of oriented cycles. Given its current state $\sigma$, we consider all the triangles that are adjacent to $\sigma$ (i.e. that share at least an edge with the cycle). The jump rate is the number of these triangles, weighted by the number of their edges common to $\sigma$. When there is a jump, say at time $t$, we chose randomly one of these triangles, say $\tau$, with a probability proportional to the number of their edges common with $\sigma$ and the Markov chain jumps from $X_{t-} = \sigma$ to $X_t = \sigma - \partial\tau$. Heuristically this deletes the common edges and replaces them with the other edges of the triangle. For example in Fig. 1(c), starting from the circle, we delete the edge adjacent to the triangle and replace it with the two other edges. The state thus remains a cycle.

More precisely the generator of this random walk is:

$$ \mathcal{A}f(\sigma) = \sum_{\tau \in S_2} (f(\sigma - \partial\tau) - f(\sigma))\langle\partial\tau, \sigma\rangle^+, \qquad (2) $$

where $x^+$ denotes the positive part of $x$ and $\langle\partial\tau, \sigma\rangle$ corresponds to the number of edges that $\tau$ and $\sigma$ have in common (with a sign corresponding to the orientation). When $f$ is a linear function, $\mathcal{A}f(\sigma) = -L_1^\uparrow f(\sigma)$ where $L_1^\uparrow$ is the up-Laplacian of order 1. The natural questions are whether the random walk is recurrent and whether we can derive diffusive scaling limits.

After recalling notions of homological and algebraic topology in Section 2, the general random walk (of order $k \ge 0$) is introduced and studied in Section 3. The case $k = 0$ is the ‘usual’ random walk, and $k = 1$ corresponds to the generator described in (2). Concerning the recurrence or transience of this Markov chain, a difficulty is that the cycle can have loops: for $k = 1$ and example (2), even if the number of vertices is finite, the state space $\mathcal{C}_1$ is infinite. It is shown in Theorem 20 that under simple conditions, the random walk is recurrent and admits a unique invariant measure. For the usual random walk that jumps from vertices to neighboring vertices, the invariant measure puts weights on vertices $u \in V$ that are proportional to their degrees. Such result is not true any more for higher dimensions.

When the simplicial complex is embedded into a geometrical space (for instance, considering the vertices in $\mathbb{R}^d$), we discuss in Section 4 the convergence of the rescaled random walk to a continuous process on a set of continuous paths. Because computations become complicated, we focus on the case of a triangulation of the torus for the sake of simplicity. We study the convergence of the random walk generator, and the limit involves the Hodge operator. The tightness is then obtained by an Aldous-Rebolledo criterion. We identify all the limiting values as solutions of the same martingale problem, but uniqueness of the solution is still left open. Finally, in Section 5, we use the invariant measure of our random walk to provide a generalization of the PageRank algorithm to the simplicial complexes which highlights the boundaries of topological structures.

# 2 Preliminaries

## 2.1 Simplicial complexes

As explained in the introduction, a natural generalization of graphs requires to consider simplicial complexes and adopt considerations from the field of homological and algebraic topology. For further reading on algebraic topology, see [3, 24, 40]. While graphs model binary relations, simplicial complexes represent higher order relations.

Given a finite or denumerable set of vertices $V$, a $k$-simplex is an unordered subset $\{v_0, v_1, \dots, v_k\}$
description is given in the paper). Recall that a simplicial complex $\mathcal{C}$ of dimension 2 is a collection of vertices $V$, of edges $E$ and of triangles $T$ such that the edges of the triangles belong to $E$ and the vertices of the edges belong to $V$. Define the oriented edge from $u$ to $v$ as $[u, v]$. The cycles are defined as chains, i.e. elements of the vector space spanned by $E^+$ the set of edges with positive orientation:

$$ \mathcal{C}_1 = \left\{ \sum_{[u,v] \in E^+} \lambda_{[u,v]}[u,v], \text{ with } \forall [u,v] \in E^+, \lambda_{[u,v]} \in \mathbb{R} \right\}, $$

with the constraint that they have no border. Defining, the boundary map $\partial$ for edges by $\partial[u, v] = v - u$, and for triangles by $\partial[u, v, w] = [v, w] - [u, w] + [u, v]$, a chain $\sigma \in \mathcal{C}_1$ is a cycle if $\partial\sigma = 0$. Our random walk $(X_t)_{t \in \mathbb{R}_+}$ is a continuous time Markov chain whose state space consists of oriented cycles. Given its current state $\sigma$, we consider all the triangles that are adjacent to $\sigma$ (i.e. that share at least an edge with the cycle). The jump rate is the number of these triangles, weighted by the number of their edges common to $\sigma$. When there is a jump, say at time $t$, we chose randomly one of these triangles, say $\tau$, with a probability proportional to the number of their edges common with $\sigma$ and the Markov chain jumps from $X_{t-} = \sigma$ to $X_t = \sigma - \partial\tau$. Heuristically this deletes the common edges and replaces them with the other edges of the triangle. For example in Fig. 1(c), starting from the circle, we delete the edge adjacent to the triangle and replace it with the two other edges. The state thus remains a cycle.

More precisely the generator of this random walk is:

$$ \mathcal{A}f(\sigma) = \sum_{\tau \in S_2} (f(\sigma - \partial\tau) - f(\sigma))\langle\partial\tau, \sigma\rangle^+, \quad (2) $$

where $x^+$ denotes the positive part of $x$ and $\langle\partial\tau, \sigma\rangle$ corresponds to the number of edges that $\tau$ and $\sigma$ have in common (with a sign corresponding to the orientation). When $f$ is a linear function, $\mathcal{A}f(\sigma) = -L_1^\uparrow f(\sigma)$ where $L_1^\uparrow$ is the up-Laplacian of order 1. The natural questions are whether the random walk is recurrent and whether we can derive diffusive scaling limits.

After recalling notions of homological and algebraic topology in Section 2, the general random walk (of order $k \ge 0$) is introduced and studied in Section 3. The case $k = 0$ is the ‘usual’ random walk, and $k = 1$ corresponds to the generator described in (2). Concerning the recurrence or transience of this Markov chain, a difficulty is that the cycle can have loops: for $k = 1$ and example (2), even if the number of vertices is finite, the state space $\mathcal{C}_1$ is infinite. It is shown in Theorem 20 that under simple conditions, the random walk is recurrent and admits a unique invariant measure. For the usual random walk that jumps from vertices to neighboring vertices, the invariant measure puts weights on vertices $u \in V$ that are proportional to their degrees. Such result is not true any more for higher dimensions.

When the simplicial complex is embedded into a geometrical space (for instance, considering the vertices in $\mathbb{R}^d$), we discuss in Section 4 the convergence of the rescaled random walk to a continuous process on a set of continuous paths. Because computations become complicated, we focus on the case of a triangulation of the torus for the sake of simplicity. We study the convergence of the random walk generator, and the limit involves the Hodge operator. The tightness is then obtained by an Aldous-Rebolledo criterion. We identify all the limiting values as solutions of the same martingale problem, but uniqueness of the solution is still left open. Finally, in Section 5, we use the invariant measure of our random walk to provide a generalization of the PageRank algorithm to the simplicial complexes which highlights the boundaries of topological structures.

# 2 Preliminaries

## 2.1 Simplicial complexes

As explained in the introduction, a natural generalization of graphs requires to consider simplicial complexes and adopt considerations from the field of homological and algebraic topology. For further reading on algebraic topology, see [3, 24, 40]. While graphs model binary relations, simplicial complexes represent higher order relations.

Given a finite or denumerable set of vertices $V$, a $k$-simplex is an unordered subset $\{v_0, v_1, \dots, v_k\}$
<figure><img src="image_1.png" /><figcaption>Figure 1: Example where simplicial complexes are the correct structure to capture the topology of the data, and in particular detect a circular structure. (a): Data drawn from a circular structure. (b): Neighborhood graph structure, that reveals two different circular structures: a triangle and a circle. (c): A simplicial complex recovers the topology of the data.</figcaption></figure>

where $v_i \in V$ and $v_i \neq v_j$ for all $i \neq j$. The faces of the $k$-simplex $\{v_0, v_1, \dots, v_k\}$ are defined as all the $(k-1)$-simplexes of the form $\{v_0, \dots, v_{j-1}, v_{j+1}, \dots, v_k\}$ with $0 \le j \le k$. The cofaces of a $k$-simplex $\tau$ are all the $(k+1)$-simplexes of which $\tau$ is a face. A simplicial complex $\mathcal{C}$ is a collection of simplexes which is closed with respect to the inclusion of faces, i.e. if $\{v_0, v_1, \dots, v_k\}$ is a $k$-simplex of $\mathcal{C}$, then all its faces are in the set of $(k-1)$-simplexes of $\mathcal{C}$. We denote by $S_k(\mathcal{C})$ the set of $k$-simplexes of $\mathcal{C}$. In the sequel, when there is no ambiguity, we will drop the dependency on $\mathcal{C}$ and simply write $S_k$. $S_k$ can be viewed as a subset of $V^{k+1}$ which itself can be embedded in $\mathbb{N}^{k+1}$. By convention, $S_0 = V$ consists of all the vertices. $S_1$ of all the edges $\{v_0, v_1\}$ of $\mathcal{C}$ linking two vertices $v_0$ and $v_1 \in V$, $v_0 \neq v_1$. $S_2, S_3$ are the set of all triangles and tetrahedra of $\mathcal{C}$ etc. Then,

$$ \mathcal{C} = \bigcup_{k \ge 0} S_k. $$

One can define an orientation on simplexes by defining an order on vertices. The oriented $k$-simplexes are denoted with square brackets, with the convention that:

$$ [v_0, \dots, v_i, \dots, v_j, \dots, v_k] = -[v_0, \dots, v_j, \dots, v_i, \dots, v_k], $$

for $0 \le i, j \le k$. Each simplex may thus appear in two species: positively or negatively oriented. We denote by $S_k^+$ (respectively $S_k^-$) the set of positively (respectively negatively) oriented $k$-simplexes containing all simplices $[v_0, \dots, v_k]$ such that $v_0 < v_1 < \dots < v_k$. For an oriented edge $[u, v]$ (going from $u$ to $v$), we call $u$ (resp. $v$) the ego (resp. alter) of the edge. Also, when $[u, v] \in S_1$, we will write $v \sim u$.

**Example 1 (Cech complex).** For $V = \{v_i, i = 1, \dots, n\}$ n points in $\mathbb{R}^d$ (or in a metric space), and $R > 0$, the Čech complex Čech($V, R$) of radius $R$ is defined as follows: $S_0 = \{v_i, i = 1, \dots, n\}$ and, for any $k > 0$, $[v_{i_0}, v_{i_1}, \dots, v_{i_k}]$ belongs to $S_k$ whenever

$$ \bigcap_{m=0}^{k} B(v_{i_m}, R) \neq \emptyset. $$

This complex has the property that its topological features, as defined below, reflect that of the geometric set $\cup_i B(v_i, R)$.

**Example 2 (Rips-Vietoris complex).** Unfortunately, the construction of the Čech complex is exponentially hard so it is very common to work with the Rips-Vietoris complex. The simplicial complex Rips($V, R$) has the same vertices $V$ and edges as the Čech complex, but for $k \ge 2$, $\{v_{i_0}, \dots, v_{i_k}\}$ belongs to $S_k$ whenever all the possible pairs made by choosing two points among $\{v_{i_0}, \dots, v_{i_k}\}$ belong to the set $S_1$ of edges of Rips($V, R$). Otherwise stated, the Rips-Vietoris is fully determined by the vertices and edges of the
<figure><img src="image_3.png" /><figcaption>Figure 1: Example where simplicial complexes are the correct structure to capture the topology of the data, and in particular detect a circular structure. (a): Data drawn from a circular structure. (b): Neighborhood graph structure, that reveals two different circular structures: a triangle and a circle. (c): A simplicial complex recovers the topology of the data.</figcaption></figure>

where $v_i \in V$ and $v_i \neq v_j$ for all $i \neq j$. The faces of the $k$-simplex $\{v_0, v_1, \dots, v_k\}$ are defined as all the $(k-1)$-simplexes of the form $\{v_0, \dots, v_{j-1}, v_{j+1}, \dots, v_k\}$ with $0 \le j \le k$. The cofaces of a $k$-simplex $\tau$ are all the $(k+1)$-simplexes of which $\tau$ is a face. A simplicial complex $\mathbf{C}$ is a collection of simplexes which is closed with respect to the inclusion of faces, i.e. if $\{v_0, v_1, \dots, v_k\}$ is a $k$-simplex of $\mathbf{C}$, then all its faces are in the set of $(k-1)$-simplexes of $\mathbf{C}$. We denote by $S_k(\mathbf{C})$ the set of $k$-simplexes of $\mathbf{C}$. In the sequel, when there is no ambiguity, we will drop the dependency on $\mathbf{C}$ and simply write $S_k$. $S_k$ can be viewed as a subset of $V^{k+1}$ which itself can be embedded in $\mathbb{N}^{k+1}$. By convention, $S_0 = V$ consists of all the vertices. $S_1$ of all the edges $\{v_0, v_1\}$ of $\mathbf{C}$ linking two vertices $v_0$ and $v_1 \in V$, $v_0 \neq v_1$. $S_2, S_3$ are the set of all triangles and tetrahedra of $\mathbf{C}$ etc. Then,

$$ \mathbf{C} = \bigcup_{k \ge 0} S_k. $$

One can define an orientation on simplexes by defining an order on vertices. The oriented $k$-simplexes are denoted with square brackets, with the convention that:

$$ [v_0, \dots, v_i, \dots, v_j, \dots, v_k] = -[v_0, \dots, v_j, \dots, v_i, \dots, v_k], $$

for $0 \le i, j \le k$. Each simplex may thus appear in two species: positively or negatively oriented. We denote by $S_k^+$ (respectively $S_k^-$) the set of positively (respectively negatively) oriented $k$-simplexes containing all simplices $[v_0, \dots, v_k]$ such that $v_0 < v_1 < \dots < v_k$. For an oriented edge $[u, v]$ (going from $u$ to $v$), we call $u$ (resp. $v$) the ego (resp. alter) of the edge. Also, when $[u, v] \in S_1$, we will write $v \sim u$.

**Example 1 (Cech complex).** For $V = \{v_i, i = 1, \dots, n\}$ n points in $\mathbb{R}^d$ (or in a metric space), and $R > 0$, the Čech complex Čech($V, R$) of radius $R$ is defined as follows: $S_0 = \{v_i, i = 1, \dots, n\}$ and, for any $k > 0$, $[v_{i_0}, v_{i_1}, \dots, v_{i_k}]$ belongs to $S_k$ whenever

$$ \bigcap_{m=0}^{k} B(v_{i_m}, R) \neq \emptyset. $$

This complex has the property that its topological features, as defined below, reflect that of the geometric set $\cup_i B(v_i, R)$.

**Example 2 (Rips-Vietoris complex).** Unfortunately, the construction of the Čech complex is exponentially hard so it is very common to work with the Rips-Vietoris complex. The simplicial complex Rips($V, R$) has the same vertices $V$ and edges as the Čech complex, but for $k \ge 2$, $\{v_{i_0}, \dots, v_{i_k}\}$ belongs to $S_k$ whenever all the possible pairs made by choosing two points among $\{v_{i_0}, \dots, v_{i_k}\}$ belong to the set $S_1$ of edges of Rips($V, R$). Otherwise stated, the Rips-Vietoris is fully determined by the vertices and edges of the
Čech complex. This graph is called the skeleton of the Rips-Vietoris. Though a priori coarser than the Čech complex, the Rips-Vietoris is not that far since (see [8])

$$Rips(V, R) \subset \check{\text{Cech}}(V, R) \subset \text{Rips}(V, 2R).$$

## 2.2 Chains and co-chains

For each integer $k$, $C_k$ is the $\mathbf{R}$-vector space spanned by the set $S_k^+$ of $k$-simplexes of $V$: an element $\sigma \in C_k$ is called a *chain* or *$k$-chain* and can be uniquely written as

$$\sigma = \sum_{\tau \in S_k^+} \lambda_{\tau}(\sigma) \tau, \qquad (3)$$

where all but a finite number of $\{\lambda_{\tau}(\sigma) \in \mathbf{R}, \tau \in S_k^+\}$ are non null. We define the support of $\sigma$ to be:

$$\text{supp } \sigma = \{\tau \in S_k^+, \lambda_{\tau}(\sigma) \neq 0\}. \qquad (4)$$

Equations (3) and (4) amount to define a scalar product that makes $C_k$ a Hilbert space and $\{\tau, \tau \in S_k^+\}$ an orthonormal basis of $C_k$. We can thus consider the canonical norm on the set $C_k$ of $k$-chains by taking

$$\|\sigma\|_{C_k}^2 = \| \sum_{\tau \in S_k^+} \lambda_{\tau}(\sigma) \tau \|_{C_k}^2 = \sum_{\tau \in S_k^+} |\lambda_{\tau}(\sigma)|^2.$$

Let $C^k$ be the topological and algebraic dual of $C_k$:

$$C^k = \{ f : C_k \to \mathbf{R}, \text{ linear and continuous} \}$$

and any element of $C^k$ is called a *cochain*. Because $C_k$ is a Hilbert space, so is $C^k$. Note that $C_k$ and $C^k$ are isomorphic and that any element $\tau \in S_k$ can be viewed either as an element of $C_k$ or as an element of $C^k$ by identification by the canonical isometries between an Hilbert space and its dual (see Example 3 below). When it will be convenient, we will indifferently manipulate chains or cochains in what follows as it is the most intuitive depending on the situation. In particular, given any $f \in C^k$ and any $\sigma \in C_k$, we will write

$$f(\sigma) = \langle f, \sigma \rangle_{C^k, C_k}.$$

Also, any function from $S_k$ to $\mathbf{R}$ can be associated canonically with a function from $C^k$ to $\mathbf{R}$.

**Example 3.** To illustrate the two assertion above, let us consider the case $k = 0$. We can identify the vertex $v \in S_0 = V \subset C_0$ to the function

$$\begin{array}{rcl}
v^* : & C_0 & \to \mathbf{R} \\
v & \mapsto & 1 \\
u & \neq & v \mapsto 0.
\end{array}$$

Then, we can extend any function $\varphi : V \to \mathbf{R}$ to a function $\varphi \in C^0$ by

$$\varphi\left(\sum_{v \in V} \lambda_v v\right) := \sum_{v \in V} \lambda_v \varphi(v).$$

## 2.3 Boundary and coboundary maps

For any integer $k$, the boundary map $\partial_k$ is the linear transformation $\partial_k : C_k \to C_{k-1}$ which acts on basis elements $[v_0, \dots, v_k] \in S_k$ as

$$\partial_k[v_0, \dots, v_k] = \sum_{i=0}^{k} (-1)^i [v_0, \dots, v_{i-1}, v_{i+1}, \dots, v_k], \qquad (5)$$
Čech complex. This graph is called the skeleton of the Rips-Vietoris. Though a priori coarser than the Čech complex, the Rips-Vietoris is not that far since (see [8])

$$Rips(V, R) \subset \check{\text{Cech}}(V, R) \subset \text{Rips}(V, 2R).$$

## 2.2 Chains and co-chains

For each integer $k$, $C_k$ is the $\mathbf{R}$-vector space spanned by the set $S_k^+$ of $k$-simplexes of $V$: an element $\sigma \in C_k$ is called a *chain* or *k-chain* and can be uniquely written as

$$\sigma = \sum_{\tau \in S_k^+} \lambda_{\tau}(\sigma) \tau, \qquad (3)$$

where all but a finite number of $\{\lambda_{\tau}(\sigma) \in \mathbf{R}, \tau \in S_k^+\}$ are non null. We define the support of $\sigma$ to be:

$$\text{supp } \sigma = \{\tau \in S_k^+, \lambda_{\tau}(\sigma) \neq 0\}. \qquad (4)$$

Equations (3) and (4) amount to define a scalar product that makes $C_k$ a Hilbert space and $\{\tau, \tau \in S_k^+\}$ an orthonormal basis of $C_k$. We can thus consider the canonical norm on the set $C_k$ of $k$-chains by taking

$$\|\sigma\|_{C_k}^2 = \| \sum_{\tau \in S_k^+} \lambda_{\tau}(\sigma) \tau \|_{C_k}^2 = \sum_{\tau \in S_k^+} |\lambda_{\tau}(\sigma)|^2.$$

Let $C^k$ be the topological and algebraic dual of $C_k$:

$$C^k = \{ f : C_k \to \mathbf{R}, \text{ linear and continuous} \}$$

and any element of $C^k$ is called a *cochain*. Because $C_k$ is a Hilbert space, so is $C^k$. Note that $C_k$ and $C^k$ are isomorphic and that any element $\tau \in S_k$ can be viewed either as an element of $C_k$ or as an element of $C^k$ by identification by the canonical isometries between an Hilbert space and its dual (see Example 3 below). When it will be convenient, we will indifferently manipulate chains or cochains in what follows as it is the most intuitive depending on the situation. In particular, given any $f \in C^k$ and any $\sigma \in C_k$, we will write

$$f(\sigma) = \langle f, \sigma \rangle_{C^k, C_k}.$$

Also, any function from $S_k$ to $\mathbf{R}$ can be associated canonically with a function from $C^k$ to $\mathbf{R}$.

**Example 3.** To illustrate the two assertion above, let us consider the case $k = 0$. We can identify the vertex $v \in S_0 = V \subset C_0$ to the function

$$\begin{array}{rcl} v^* : & C_0 & \to \mathbf{R} \\ & v & \mapsto 1 \\ & u \neq v & \mapsto 0. \end{array}$$

Then, we can extend any function $\varphi : V \to \mathbf{R}$ to a function $\varphi \in C^0$ by

$$\varphi\left(\sum_{v \in V} \lambda_v v\right) := \sum_{v \in V} \lambda_v \varphi(v).$$

## 2.3 Boundary and coboundary maps

For any integer $k$, the boundary map $\partial_k$ is the linear transformation $\partial_k : C_k \to C_{k-1}$ which acts on basis elements $[v_0, \dots, v_k] \in S_k$ as

$$\partial_k[v_0, \dots, v_k] = \sum_{i=0}^{k} (-1)^i [v_0, \dots, v_{i-1}, v_{i+1}, \dots, v_k], \qquad (5)$$
<figure><img src="image_2.png" /><figcaption>Table 1: Examples of boundary maps. From left to right. An application over 1-simplexes. Over a 2-simplex. Over a 3-simplex, turning a filled tetrahedron to an empty one.</figcaption></figure>

and $\partial_0$ is the null function. Examples of such operations are given in Table 1. The maps $(\partial_k, k \ge 1)$ link the spaces $C_k$'s as follows:

$$ \dots \xrightarrow{\partial_{k+2}} C_{k+1} \xrightarrow{\partial_{k+1}} C_k \xrightarrow{\partial_k} \dots \xrightarrow{\partial_2} C_1 \xrightarrow{\partial_1} C_0. \qquad (6) $$

It can then easily checked that for any integer $k$,

$$ \partial_k \circ \partial_{k+1} = 0. \qquad (7) $$

In topology, a sequence of vector spaces and linear transformations satisfying (6) and (7) is called a chain complex. Equation (7) implies that $B_k = \text{im} \partial_{k+1} \subset \text{ker} \partial_k = Z_k$, for $k \ge 0$, and we can define the $k$-th homology vector space $H_k$ as the quotient vector space,

$$ H_k = \ker \partial_k / \text{im} \partial_{k+1}. \qquad (8) $$

The $k$-th Betti number of $C$ is defined as its dimension:

$$ \beta_k = \dim H_k = \dim(\ker \partial_k) - \dim(\text{im} \partial_{k+1}). \qquad (9) $$

Notice that an element of $C_1$ is a sum of edges. It belongs to $\ker \partial_1$ whenever these oriented edges form a cycle, in the sense of graph theory. So elements of $\ker \partial_k$ are called $k$-cycles.

The Fig. 2.3 illustrate the chain complex described above.

<figure><img src="image_3.png" /><figcaption>Figure 2: A chain complex showing the sets $C_k$, $Z_k$ and $B_k$.</figcaption></figure>

**Example 4.** Let us consider the case of $k = 0$ again to illustrate (9). Since $\partial_0 \equiv 0$, we have $\ker \partial_0 = V$. Also, we have $\text{im} \partial_1 = \{u - v, [u, v] \in S_1\}$. Hence, $H_0 = \text{span}(V)/\{u - v, [u, v] \in S_1\}$ consists of equivalence classes of vertices which can be linked by a path in the graph. We thus recover that $\beta_0$ is the number of connected components of the graph. Recall that the latter number also corresponds to the number of zeros in the spectrum of the graph Laplacian (1).
<figure><img src="image_4.png" /><figcaption>Table 1: Examples of boundary maps. From left to right. An application over 1-simplexes. Over a 2-simplex. Over a 3-simplex, turning a filled tetrahedron to an empty one.</figcaption></figure>

and $\partial_0$ is the null function. Examples of such operations are given in Table 1. The maps $(\partial_k, k \ge 1)$ link the spaces $C_k$'s as follows:

$$ \dots \xrightarrow{\partial_{k+2}} C_{k+1} \xrightarrow{\partial_{k+1}} C_k \xrightarrow{\partial_k} \dots \xrightarrow{\partial_2} C_1 \xrightarrow{\partial_1} C_0. \qquad (6) $$

It can then easily checked that for any integer $k$,

$$ \partial_k \circ \partial_{k+1} = 0. \qquad (7) $$

In topology, a sequence of vector spaces and linear transformations satisfying (6) and (7) is called a chain complex. Equation (7) implies that $B_k = \text{im } \partial_{k+1} \subset \text{ker } \partial_k = Z_k$, for $k \ge 0$, and we can define the $k$-th homology vector space $H_k$ as the quotient vector space,

$$ H_k = \ker \partial_k / \text{im } \partial_{k+1}. \qquad (8) $$

The $k$-th Betti number of $C$ is defined as its dimension:

$$ \beta_k = \dim H_k = \dim(\ker \partial_k) - \dim(\text{im } \partial_{k+1}). \qquad (9) $$

Notice that an element of $C_1$ is a sum of edges. It belongs to $\ker \partial_1$ whenever these oriented edges form a cycle, in the sense of graph theory. So elements of $\ker \partial_k$ are called $k$-cycles.

The Fig. 2.3 illustrate the chain complex described above.

<figure><img src="image_5.png" /><figcaption>Figure 2: A chain complex showing the sets $C_k$, $Z_k$ and $B_k$.</figcaption></figure>

**Example 4.** Let us consider the case of $k = 0$ again to illustrate (9). Since $\partial_0 \equiv 0$, we have $\ker \partial_0 = V$. Also, we have $\text{im } \partial_1 = \{u - v, [u, v] \in S_1\}$. Hence, $H_0 = \text{span}(V)/\{u - v, [u, v] \in S_1\}$ consists of equivalence classes of vertices which can be linked by a path in the graph. We thus recover that $\beta_0$ is the number of connected components of the graph. Recall that the latter number also corresponds to the number of zeros in the spectrum of the graph Laplacian (1).
**Remark 5.** An element of $\ker \partial_1$ which is not in $\im \partial_2$ is a cycle which cannot be written as the boundary of a sum of triangles in $S_2$. For instance, if

$$S_1 = \{[ab], [bc], [cd], [ad]\} \text{ and } S_2 = \emptyset,$$

then the cycle

$$[ab] + [bc] + [cd] - [ad]$$

which corresponds to the edges of a 4-gone, cannot be written as a sum of triangles since $S_2$ is empty. Thus $\beta_1 = 1$ in this case.

As $C_k$ and $C^k$ are Hilbert spaces, we can define the coboundary map $\partial_k^* : C^{k-1} \to C^k$ as the adjoint of $\partial_k$: namely for $f \in C^{k-1}$, $\partial_k^* f \in C^k$ is defined by its action over a $k$-chain by

$$(\partial_k^* f)[v_0, \dots, v_k] = \sum_{i=0}^{k} (-1)^i \langle f, [v_0, \dots, v_{i-1}, v_{i+1}, \dots, v_k] \rangle_{C^{k-1}, C_{k-1}} = f(\partial_k[v_0, \dots, v_k]). \quad (10)$$

We will set by convention $\partial_0^* = 0$.

**Remark 6** (Interpretation of the coboundary map in the case $k=1$). Recall that $C_0$ is generated as a $\mathbf{R}$-vector space by the points $v \in V$. Let us denote by $\{v^*, v \in V\}$ the corresponding dual basis of $C^0$: $v^*(v) = 1$ and $v^*(w) = 0$ for $w \neq v$. Hence, following (10), we have for any function $f \in C^0$,

$$\partial_1^* f[v_0, v_1] = f(v_1) - f(v_0).$$

In particular, if $f = w^*$ for $w \in V$,

$$(\partial_1^* w^*)([v_0, v_1]) = -\langle w^*, v_0 \rangle_{C^0, C_0} + \langle w^*, v_1 \rangle_{C^0, C_0} = \begin{cases} 0 & \text{if } w \neq v_0, v_1 \\ -1 & \text{if } w = v_0 \\ 1 & \text{if } w = v_1. \end{cases}$$

The above computation shows that the coboundary of a vertex $w$ gives a weight 1 (resp. -1) to oriented edges having $w$ as alter (resp. ego), i.e. arriving at (resp. departing from) $w$. The coboundary map can then be interpreted in terms of fluxes.

**Example 7.** Let us consider an example with four vertices: a, b, c and d. Consider that the edge $[a, b]$ belongs to the two triangles $[a, b, c]$ and $[a, b, d]$. Locally, the matrix representation of $\partial_2$ looks like

$$\begin{bmatrix} [a, b, c] & [a, b, d] \\ [a, b] & 1 \\ [a, c] & -1 \\ [a, d] & 0 \\ [b, c] & 1 \\ [b, d] & 0 \end{bmatrix}.$$

The matrix representation of $\partial_2^*$ is of course the transposed of the matrix representing $\partial_2$ and recalling that we have identified $C_1$ (resp. $C_2$) to its dual $C^1$ (resp $C^2$),

$$\partial_2^*[a, b] = [a, b, c] + [a, b, d] = \sum_{\tau \in S_2^+} \langle [a, b]^*, \partial_2 \tau \rangle_{C^1, C_1} \tau. \quad (11)$$

The bracket on the right hand side counts the occurrence of the edge $[a, b]$ among the faces of $\tau$. Otherwise stated, the coboundary of an edge is the sum of the triangles which contain it, respecting its orientation.

As before, the $k$-th cohomology vector space, denoted by $H^k$, is defined as

$$H^k = \ker \partial_k^* / \im \partial_{k-1}^* \quad (12)$$

and is the dual of $H_k$.
**Remark 5.** An element of $\ker \partial_1$ which is not in $\im \partial_2$ is a cycle which cannot be written as the boundary of a sum of triangles in $S_2$.
For instance, if

$$S_1 = \{[ab], [bc], [cd], [ad]\} \text{ and } S_2 = \emptyset,$$

then the cycle

$$[ab] + [bc] + [cd] - [ad]$$

which corresponds to the edges of a 4-gone, cannot be written as a sum of triangles since $S_2$ is empty.
Thus $\beta_1 = 1$ in this case.

As $C_k$ and $C^k$ are Hilbert spaces, we can define the coboundary map $\partial_k^* : C^{k-1} \to C^k$ as the adjoint of $\partial_k$: namely for $f \in C^{k-1}$, $\partial_k^* f \in C^k$ is defined by its action over a $k$-chain by

$$(\partial_k^* f)[v_0, \dots, v_k] = \sum_{i=0}^{k} (-1)^i \langle f, [v_0, \dots, v_{i-1}, v_{i+1}, \dots, v_k] \rangle_{C^{k-1}, C_{k-1}} = f(\partial_k[v_0, \dots, v_k]). \quad (10)$$

We will set by convention $\partial_0^* = 0$.

**Remark 6** (Interpretation of the coboundary map in the case $k=1$). Recall that $C_0$ is generated as a $\mathbf{R}$-vector space by the points $v \in V$. Let us denote by $\{v^*, v \in V\}$ the corresponding dual basis of $C^0$: $v^*(v) = 1$ and $v^*(w) = 0$ for $w \neq v$. Hence, following (10), we have for any function $f \in C^0$,

$$\partial_1^* f[v_0, v_1] = f(v_1) - f(v_0).$$

In particular, if $f = w^*$ for $w \in V$,

$$(\partial_1^* w^*)([v_0, v_1]) = -\langle w^*, v_0 \rangle_{C^0, C_0} + \langle w^*, v_1 \rangle_{C^0, C_0} = \begin{cases} 0 & \text{if } w \neq v_0, v_1 \\ -1 & \text{if } w = v_0 \\ 1 & \text{if } w = v_1. \end{cases}$$

The above computation shows that the coboundary of a vertex $w$ gives a weight 1 (resp. -1) to oriented edges having $w$ as alter (resp. ego), i.e. arriving at (resp. departing from) $w$. The coboundary map can then be interpreted in terms of fluxes.

**Example 7.** Let us consider an example with four vertices: a, b, c and d. Consider that the edge $[a, b]$ belongs to the two triangles $[a, b, c]$ and $[a, b, d]$. Locally, the matrix representation of $\partial_2$ looks like

$$\begin{bmatrix} [a, b, c] & [a, b, d] \\ [a, b] & 1 \\ [a, c] & -1 \\ [a, d] & 0 \\ [b, c] & 1 \\ [b, d] & 0 \end{bmatrix} \begin{array}{l} \langle [a, b]^*, \partial_2 \tau \rangle_{C^1, C_1} \tau. \end{array}$$

The matrix representation of $\partial_2^*$ is of course the transposed of the matrix representing $\partial_2$ and recalling that we have identified $C_1$ (resp. $C_2$) to its dual $C^1$ (resp $C^2$),

$$\partial_2^*[a, b] = [a, b, c] + [a, b, d] = \sum_{\tau \in S_2^+} \langle [a, b]^*, \partial_2 \tau \rangle_{C^1, C_1} \tau. \quad (11)$$

The bracket on the right hand side counts the occurrence of the edge $[a, b]$ among the faces of $\tau$. Otherwise stated, the coboundary of an edge is the sum of the triangles which contain it, respecting its orientation.

As before, the $k$-th cohomology vector space, denoted by $H^k$, is defined as

$$H^k = \ker \partial_k^* / \im \partial_{k-1}^* \quad (12)$$

and is the dual of $H_k$.
## 2.4 Combinatorial Laplacian

A crucial notion for the following is that of combinatorial Laplacian (see [19, 38] for details). We know that

$$ \begin{array}{ccc} \xrightarrow{\partial_{k+2}} & C_{k+1} & \xrightarrow{\partial_{k+1}} C_k \\ \xleftarrow{\partial_{k+1}} & \xrightarrow{\partial_{k+1}} & \xleftarrow{\partial_k} C_{k-1} \\ \xrightarrow{\partial_{k+2}} & C^{k+1} & \xrightarrow{\partial_k} C^{k-1} \\ \xleftarrow{\partial_{k+1}} & \xleftarrow{\partial_{k+1}} & \xleftarrow{\partial_{k-1}} C^{k-1} \\ \xleftarrow{\partial_{k+1}} & \xleftarrow{\partial_{k-1}} & \xleftarrow{\partial_{k-1}} C^{k-1} \end{array} $$

where the two tips arrows mean that we have an isometric isomorphism between the two concerned Hilbert spaces. Since we have identified the spaces $C_k$ and $C^k$ for any $k \in \mathbb{N}$, we may consider

$$ L_k^\uparrow := \partial_{k+1} \partial_{k+1}^* : C_k \xrightarrow{\partial_{k+1}^*} C_{k+1} \xrightarrow{\partial_{k+1}} C_k \quad (13) $$

and

$$ L_k^\downarrow := \partial_k^* \partial_k : C_k \xrightarrow{\partial_k} C_{k-1} \xrightarrow{\partial_k^*} C_k. \quad (14) $$

These latter operators are called respectively the upper and lower Laplacians (of order $k$). The combinatorial Laplacian of order $k$ is defined as

$$ \begin{array}{l} L_k : C_k \longrightarrow C_k \\ \sigma \longmapsto (\partial_{k+1} \partial_{k+1}^* + \partial_k^* \partial_k) (\sigma) = L_k^\uparrow(\sigma) + L_k^\downarrow(\sigma). \end{array} \quad (15) $$

By definition, all these three operators are self-adjoint, non-negative and compact. Their eigenvalues are thus real and non-negative. Furthermore, the combinatorial Hodge theorem [20] says that

**Theorem 8.** For any $k \in \mathbb{N}$, we have

$$ C_k = \operatorname{im} \partial_{k+1} \oplus \operatorname{im} \partial_k^* \oplus \operatorname{ker} L_k. \quad (16) $$

It follows that

$$ \ker L_k \approx H_k. \quad (17) $$

In particular, the $k$-th Betti number is the dimension of the null space of $L_k$:

$$ \beta_k = \dim \ker(L_k). \quad (18) $$

**Remark 9.** The combinatorial Laplacian of order 0 corresponds to the graph Laplacian. Since we set by convention $\partial_0 = 0$, $L_0 = L_0^\uparrow = \partial_1 \partial_1^*$. The map $\partial_1$ maps edges to vertices and its matrix representation is exactly the so-called incidence matrix $B$ of the graph $(V, S_1)$: for $V = \{v_0, \dots, v_k\}$ and for $(e_j)$ an enumeration of the set of oriented edges $S_1^+$,

$$ B_{ij} = \begin{cases} 1 & \text{if } v_i \text{ is the ego of } e_j \\ -1 & \text{if } v_i \text{ is the alter of } e_j \\ 0 & \text{otherwise.} \end{cases} $$

Thus, $L_0 = BB^t$ is such that

$$ (L_0)_{ij} = \begin{cases} \deg(v_i) & \text{if } i = j, \\ -1 & \text{if } v_i \text{ is adjacent to } v_j, \\ 0 & \text{otherwise.} \end{cases} $$

The map $L_0$ from $C_0$ into itself is characterized for $v \in V$ by:

$$ L_0 v^* = - \sum_{w \in V : [vw] \in S_1} (w^* - v^*) = \sum_{w \in V : [vw] \in S_1} \partial_1 [vw]. $$
## 2.4 Combinatorial Laplacian

A crucial notion for the following is that of combinatorial Laplacian (see [19, 38] for details). We know that

$$ \begin{array}{ccc} \xrightarrow{\partial_{k+2}} & C_{k+1} \xrightarrow{\partial_{k+1}} & C_k \xrightarrow{\partial_k} C_{k-1} \xrightarrow{\partial_{k-1}} \\ \xleftarrow{\partial_{k+2}} & C^{k+1} \xleftarrow{\partial_{k+1}^*} & C^k \xleftarrow{\partial_k^*} C^{k-1} \xleftarrow{\partial_{k-1}^*} \end{array} $$

where the two tips arrows mean that we have an isometric isomorphism between the two concerned Hilbert spaces. Since we have identified the spaces $C_k$ and $C^k$ for any $k \in \mathbb{N}$, we may consider

$$ L_k^\uparrow := \partial_{k+1} \partial_{k+1}^* : C_k \xrightarrow{\partial_{k+1}^*} C_{k+1} \xrightarrow{\partial_{k+1}} C_k \quad (13) $$

and

$$ L_k^\downarrow := \partial_k^* \partial_k : C_k \xrightarrow{\partial_k} C_{k-1} \xrightarrow{\partial_k^*} C_k. \quad (14) $$

These latter operators are called respectively the upper and lower Laplacians (of order $k$). The combinatorial Laplacian of order $k$ is defined as

$$ \begin{array}{l} L_k : C_k \to C_k \\ \sigma \mapsto (\partial_{k+1} \partial_{k+1}^* + \partial_k^* \partial_k) (\sigma) = L_k^\uparrow(\sigma) + L_k^\downarrow(\sigma). \end{array} \quad (15) $$

By definition, all these three operators are self-adjoint, non-negative and compact. Their eigenvalues are thus real and non-negative. Furthermore, the combinatorial Hodge theorem [20] says that

**Theorem 8.** For any $k \in \mathbb{N}$, we have

$$ C_k = \mathrm{im} \partial_{k+1} \oplus \mathrm{im} \partial_k^* \oplus \ker L_k. \quad (16) $$

It follows that

$$ \ker L_k \simeq H_k. \quad (17) $$

In particular, the $k$-th Betti number is the dimension of the null space of $L_k$:

$$ \beta_k = \dim \ker(L_k). \quad (18) $$

**Remark 9.** The combinatorial Laplacian of order 0 corresponds to the graph Laplacian. Since we set by convention $\partial_0 = 0$, $L_0 = L_0^\uparrow = \partial_1 \partial_1^*$. The map $\partial_1$ maps edges to vertices and its matrix representation is exactly the so-called incidence matrix $B$ of the graph $(V, S_1)$: for $V = \{v_0, \cdots, v_k\}$ and for $(e_j)$ an enumeration of the set of oriented edges $S_1^+$,

$$ B_{ij} = \begin{cases} 1 & \text{if } v_i \text{ is the ego of } e_j \\ -1 & \text{if } v_i \text{ is the alter of } e_j \\ 0 & \text{otherwise.} \end{cases} $$

Thus, $L_0 = BB^t$ is such that

$$ (L_0)_{ij} = \begin{cases} \deg(v_i) & \text{if } i = j, \\ -1 & \text{if } v_i \text{ is adjacent to } v_j, \\ 0 & \text{otherwise.} \end{cases} $$

The map $L_0$ from $C_0$ into itself is characterized for $v \in V$ by:

$$ L_0 v^* = - \sum_{w \in V : [vw] \in S_1} (w^* - v^*) = \sum_{w \in V : [vw] \in S_1} \partial_1 [vw]. $$
$$
\begin{align*}
\text{Thus, for a function } f = \sum_{v \in V} \lambda_v v^* \in \mathcal{C}^0, \\
& -L_0 f(u) = - \sum_{v \in V} \lambda_v L_0 v^*(u) \\
& = \sum_{v \in V} \lambda_v \sum_{w \in V : [vw] \in \mathcal{S}_1} (w^* - v^*)(u) \\
& = \sum_{v \sim u} \lambda_v - \lambda_u \mathrm{Card}(w \sim u) \\
& = \sum_{v \sim u} (f(v) - f(u)) = \mathcal{A}f(u). \tag{19}
\end{align*}
$$

Thus as mentioned above, $-L_0$ appears as the generator $\mathcal{A}$ of the continuous time random walk on the graph $(V, \mathcal{S}_1)$ as defined in (1).

To describe $L_k$, we need to introduce the notion of lower and upper adjacency for $k$-simplexes. Two $k$-simplexes are said to be upper adjacent whenever they are two faces of a common $k+1$ simplex of $\mathcal{C}$. Two $k$-simplexes are said to be lower adjacent whenever they are cofaces of a common $k-1$ simplex. For a simplex $\tau$, its upper degree, $\deg_{\uparrow}(\tau)$, is the number of simplexes which are upper adjacent to it. Two upper adjacent simplexes are said to be similarly oriented if the orientation they would inherit from their common higher order simplex coincides with their orientation. They are said to be dissimilarly oriented otherwise. The lower degree $\deg_{\downarrow}(\tau)$ is the number of simplexes which are lower adjacent to $\tau$. Two lower adjacent simplexes are similarly oriented whenever they induce the same orientation to their intersection.

**Example 10.** For instance, two edges are upper adjacent if they are part of a common triangle and they are lower adjacent if they share a vertex.

The lower and upper adjacency matrix are defined as it can be expected. For $\tau$ and $\tau' \in S_k$,

$$
A_k^{\uparrow/\downarrow}(\tau, \tau') = \begin{cases} 1 & \text{if } \tau \text{ and } \tau' \text{ are upper/lower adjacent and similarly oriented,} \\ -1 & \text{if } \tau \text{ and } \tau' \text{ are upper/lower adjacent and dissimilarly oriented,} \\ 0 & \text{otherwise.} \end{cases}
$$

Consider also $D_k$ the diagonal matrix whose entries are the upper degrees of each $k$-simplex.

We can compute the matrix of the combinatorial Laplacian of order $k$ (15). Then (see [38]),

$$
L_k = L_k^{\uparrow} + L_k^{\downarrow} = (D_k - A_k^{\uparrow}) + ((k+1) \mathrm{Id} + A_k^{\downarrow}).
$$

The map $L_k^{\uparrow}$ has the features of a generator of a Markov process. Indeed, considering back the computation in Remark 9, we see that the upper Laplacian can be explained in terms of upper-adjacency. For a vertex $u$ ($k=0$), we consider all the edges that are upper-adjacent to $u$, choose one with $u$ as ego and jump to the alter of this edge. This can be generalized for larger orders $k \ge 0$: we consider all the $k+1$ simplexes that are adjacent to a given $k$ simplex $\tau$, choose one at random that will determine the next movement (this shall be precised in the sequel). Associating to $L_k^{\uparrow}$ a Markov chain provides a probabilistic interpretation to this operator which can help understand it better. In Parzanchevski and Rosenthal [43], a connection between this random walk $Y$ and homology is made by considering the ‘expectation process’ defined for an oriented edge $e \in \mathcal{S}_1$ by $\mathcal{E}_t(e) = \mathbf{P}(Y_t = e) - \mathbf{P}(Y_t = -e)$.

As to the map $L_k^{\downarrow}$, Mukherjee and Steenbergen [39] proposed a similar random walk exploiting the lower-
adjacency, but the generator $L_k^{\downarrow}$ does not correspond to the generator of a Markov process. These authors
introduce killings to deal with this problem.

However, the following result says that as long as we are concerned with spectral properties, we can
retrieve the information about $L_k^{\downarrow}$ by looking at $L_{k-1}^{\uparrow}$ (see [62]):

**Theorem 11.** Let $\lambda > 0$ and $f$ be an eigenvector of $L_{k-1}^{\uparrow}$. Then, $\partial_k^* f$ is a $\lambda$-eigenvector of $L_k^{\downarrow}$. Conversely, if $g$ is a $\lambda$-eigenvector of $L_k^{\downarrow}$, then $\partial_k g$ is a $\lambda$-eigenvector of $L_{k-1}^{\uparrow}$.
$$
\begin{align*}
\text{Thus, for a function } f = \sum_{v \in V} \lambda_v v^* \in \mathcal{C}^0, \\
& -L_0 f(u) = - \sum_{v \in V} \lambda_v L_0 v^*(u) \\
& = \sum_{v \in V} \lambda_v \sum_{w \in V : [vw] \in \mathcal{S}_1} (w^* - v^*)(u) \\
& = \sum_{v \sim u} \lambda_v - \lambda_u \mathrm{Card}(w \sim u) \\
& = \sum_{v \sim u} (f(v) - f(u)) = \mathcal{A}f(u). \tag{19}
\end{align*}
$$

Thus as mentioned above, $-L_0$ appears as the generator $\mathcal{A}$ of the continuous time random walk on the graph $(V, \mathcal{S}_1)$ as defined in (1).

To describe $L_k$, we need to introduce the notion of lower and upper adjacency for $k$-simplexes. Two $k$-simplexes are said to be upper adjacent whenever they are two faces of a common $k+1$ simplex of $\mathcal{C}$. Two $k$-simplexes are said to be lower adjacent whenever they are cofaces of a common $k-1$ simplex. For a simplex $\tau$, its upper degree, $\deg_{\uparrow}(\tau)$, is the number of simplexes which are upper adjacent to it. Two upper adjacent simplexes are said to be similarly oriented if the orientation they would inherit from their common higher order simplex coincides with their orientation. They are said to be dissimilarly oriented otherwise. The lower degree $\deg_{\downarrow}(\tau)$ is the number of simplexes which are lower adjacent to $\tau$. Two lower adjacent simplexes are similarly oriented whenever they induce the same orientation to their intersection.

**Example 10.** For instance, two edges are upper adjacent if they are part of a common triangle and they are lower adjacent if they share a vertex.

The lower and upper adjacency matrix are defined as it can be expected. For $\tau$ and $\tau' \in S_k$,

$$
A_k^{\uparrow/\downarrow}(\tau, \tau') = \begin{cases} 1 & \text{if } \tau \text{ and } \tau' \text{ are upper/lower adjacent and similarly oriented,} \\ -1 & \text{if } \tau \text{ and } \tau' \text{ are upper/lower adjacent and dissimilarly oriented,} \\ 0 & \text{otherwise.} \end{cases}
$$

Consider also $D_k$ the diagonal matrix whose entries are the upper degrees of each $k$-simplex.

We can compute the matrix of the combinatorial Laplacian of order $k$ (15). Then (see [38]),

$$
L_k = L_k^{\uparrow} + L_k^{\downarrow} = (D_k - A_k^{\uparrow}) + ((k+1) \mathrm{Id} + A_k^{\downarrow}).
$$

The map $L_k^{\uparrow}$ has the features of a generator of a Markov process. Indeed, considering back the computation in Remark 9, we see that the upper Laplacian can be explained in terms of upper-adjacency. For a vertex $u$ ($k=0$), we consider all the edges that are upper-adjacent to $u$, choose one with $u$ as ego and jump to the alter of this edge. This can be generalized for larger orders $k \ge 0$: we consider all the $k+1$ simplexes that are adjacent to a given $k$ simplex $\tau$, choose one at random that will determine the next movement (this shall be precised in the sequel). Associating to $L_k^{\uparrow}$ a Markov chain provides a probabilistic interpretation to this operator which can help understand it better. In Parzanchevski and Rosenthal [43], a connection between this random walk $Y$ and homology is made by considering the ‘expectation process’ defined for an oriented edge $e \in \mathcal{S}_1$ by $\mathcal{E}_t(e) = \mathbf{P}(Y_t = e) - \mathbf{P}(Y_t = -e)$.

As to the map $L_k^{\downarrow}$, Mukherjee and Steenbergen [39] proposed a similar random walk exploiting the lower-
adjacency, but the generator $L_k^{\downarrow}$ does not correspond to the generator of a Markov process. These authors
introduce killings to deal with this problem.

However, the following result says that as long as we are concerned with spectral properties, we can
retrieve the information about $L_k^{\downarrow}$ by looking at $L_{k-1}^{\uparrow}$ (see [62]):

**Theorem 11.** Let $\lambda > 0$ and $f$ be an eigenvector of $L_{k-1}^{\uparrow}$. Then, $\partial_k^* f$ is a $\lambda$-eigenvector of $L_k^{\downarrow}$. Conversely, if $g$ is a $\lambda$-eigenvector of $L_k^{\downarrow}$, then $\partial_k g$ is a $\lambda$-eigenvector of $L_{k-1}^{\uparrow}$.
Proof. If $f$ satisfies $L_{k-1}^{\uparrow}f = \lambda f$, then

$$L_k^{\downarrow}(\partial_k^* f) = \partial_k^* \partial_k \partial_k^* f = \partial_k^* L_{k-1}^{\uparrow} f = \lambda \partial_k^* f.$$

The proof is similar for the converse.

This means that it is reasonable to look only at the maps $L_k^{\uparrow}$ as long as we are interested in properties related with the spectral decomposition of these operators.

## 3 Random walk

The idea behind the dynamics of our random walk is the following. The usual random walk on a graph goes from vertex to vertex. The generator of the continuous time random walk can be written as

$$Af(u) = \sum_{v \sim u} f(v) - f(u) = \sum_{v \in V : [uv] \in S_1} f(u + \partial_1[uv]) - f(u). \quad (20)$$

In the next dimension, $k = 1$, points are replaced by edges and edges by triangles. If we follow Parzanchevski and Rosenthal [43, 44] or Kaufman and Oppenheim [32], a natural edge-valued random walk consists in jumping from the current edge $e$ to a uniformly chosen upper-adjacent edge. These 'higher-order' random walks can be extended to $k \ge 1$. Mukherjee and Steenbergen proposed a similar random walks exploiting the lower-adjacency. But if we look for an analogue of (20), another way is to add the boundary of a triangle to an edge, which gives a combination of edges, i.e. a 2-chain. It is thus natural not to restrict to edges, but to consider a random walk that takes its values in $C_2$ or more generally $C_k$.

Recall that in (8), the homology space $H_k = \ker \partial_k / \operatorname{im} \partial_{k+1}$, a natural way to explore the homology classes of $H_k$ is to start with an element of $\ker \partial_k$ and then have transitions in $\operatorname{im} \partial_{k+1}$. Proceeding so, the random walk will remain in the homology class of its initial element, in the same way as the usual random walk remained in the connected component of its initial condition. Let us now describe the transitions and generator of this random walk.

### 3.1 Generator of the chain-valued random-walk

In what follows, $k \ge 1$ is fixed.

**Space of test functions.** To define the generator of the random walk, we first introduce a space of functions on which it will operate.

**Definition 12.** Consider $\mathcal{D}$ the space of functions from $C_k$ to $\mathbb{R}$ of the form

$$F(\sigma) = f(\langle \eta_1, \sigma \rangle_{C^k, C_k}, \dots, \langle \eta_m, \sigma \rangle_{C^k, C_k}) \quad (21)$$

for some $m \ge 1$, $(\eta_1, \dots, \eta_m)$ some elements of $C^k$ and $f$ measurable and bounded from $\mathbb{R}^m$ into $\mathbb{R}$.

We define the support of $F$ as:

$$\text{supp } F = \bigcup_{i=1}^{m} \text{supp } \eta_i,$$

where we recall that $\text{supp } \eta_i$ is defined in (4).

**Lemma 13.** The space $\mathcal{D}$ is separating in $B(C_k)$, the Banach space of bounded measurable functions from $C_k$ to $\mathbb{R}$, equipped with the sup-norm.

Proof. Since $C^k$ can be embedded as a closed subset of $l^2(\mathbb{N}^{k+1})$, it is a separable Hilbert space, we can consider a dense sequence $(\eta_n, n \ge 1)$ and

$$F_m = \sigma\{\langle \eta_i, \cdot \rangle_{C^k, C_k}, i = 1, \dots, m\}.$$

Since $C_k$ is an Hilbert space, its Borel $\sigma$-field is equal to $\bigvee_m F_m$, hence the result.
Proof. If $f$ satisfies $L_{k-1}^{\uparrow}f = \lambda f$, then

$$L_k^{\downarrow}(\partial_k^* f) = \partial_k^* \partial_k \partial_k^* f = \partial_k^* L_{k-1}^{\uparrow} f = \lambda \partial_k^* f.$$

The proof is similar for the converse.

This means that it is reasonable to look only at the maps $L_k^{\uparrow}$ as long as we are interested in properties related with the spectral decomposition of these operators.

## 3 Random walk

The idea behind the dynamics of our random walk is the following. The usual random walk on a graph goes from vertex to vertex. The generator of the continuous time random walk can be written as

$$Af(u) = \sum_{v \sim u} f(v) - f(u) = \sum_{v \in V : [uv] \in S_1} f(u + \partial_1[uv]) - f(u). \quad (20)$$

In the next dimension, $k = 1$, points are replaced by edges and edges by triangles. If we follow Parzanchevski and Rosenthal [43, 44] or Kaufman and Oppenheim [32], a natural edge-valued random walk consists in jumping from the current edge $e$ to a uniformly chosen upper-adjacent edge. These 'higher-order' random walks can be extended to $k \ge 1$. Mukherjee and Steenbergen proposed a similar random walks exploiting the lower-adjacency. But if we look for an analogue of (20), another way is to add the boundary of a triangle to an edge, which gives a combination of edges, i.e. a 2-chain. It is thus natural not to restrict to edges, but to consider a random walk that takes its values in $C_2$ or more generally $C_k$.

Recall that in (8), the homology space $H_k = \ker \partial_k / \text{im} \partial_{k+1}$, a natural way to explore the homology classes of $H_k$ is to start with an element of $\ker \partial_k$ and then have transitions in $\text{im} \partial_{k+1}$. Proceeding so, the random walk will remain in the homology class of its initial element, in the same way as the usual random walk remained in the connected component of its initial condition. Let us now describe the transitions and generator of this random walk.

### 3.1 Generator of the chain-valued random-walk

In what follows, $k \ge 1$ is fixed.

**Space of test functions.** To define the generator of the random walk, we first introduce a space of functions on which it will operate.

**Definition 12.** Consider $\mathcal{D}$ the space of functions from $C_k$ to $\mathbb{R}$ of the form

$$F(\sigma) = f(\langle \eta_1, \sigma \rangle_{C^k, C_k}, \dots, \langle \eta_m, \sigma \rangle_{C^k, C_k}) \quad (21)$$

for some $m \ge 1$, $(\eta_1, \dots, \eta_m)$ some elements of $C^k$ and $f$ measurable and bounded from $\mathbb{R}^m$ into $\mathbb{R}$.

We define the support of $F$ as:

$$\text{supp } F = \bigcup_{i=1}^{m} \text{supp } \eta_i,$$

where we recall that $\text{supp } \eta_i$ is defined in (4).

**Lemma 13.** The space $\mathcal{D}$ is separating in $B(C_k)$, the Banach space of bounded measurable functions from $C_k$ to $\mathbb{R}$, equipped with the sup-norm.

Proof. Since $C^k$ can be embedded as a closed subset of $l^2(\mathbb{N}^{k+1})$, it is a separable Hilbert space, we can consider a dense sequence $(\eta_n, n \ge 1)$ and

$$F_m = \sigma\{\langle \eta_i, \cdot \rangle_{C^k, C_k}, i = 1, \dots, m\}.$$

Since $C_k$ is an Hilbert space, its Borel $\sigma$-field is equal to $\bigvee_m F_m$, hence the result.
**Transition kernel.** Let us explain the transition of our chain-valued random walk. For the sake of simplicity, imagine here that $k = 1$. Assume that we are in a state $\sigma \in C_1$ (a cycle for $k = 1$). Because transitions are in $\text{im } \partial_2$, let us consider an element of this space, say $\partial_2\tau$ for $\tau \in S_2$ (a triangle). $\partial_2\tau$ defines a possible transition if $\tau$ is upper-adjacent to $\sigma$, i.e. if $\tau$ and $\sigma$ share at least one edge. All the possible transitions from $\sigma$ are obtained by letting $\tau$ vary in $S_2$. The more $\tau$ and $\sigma$ have edges in common and the more $\tau$ will be likely to define the next step of the random walk and we thus need to define a weight to account for this.

For a k-chain $\sigma \neq 0$ and an oriented $(k+1)$-simplex $\tau \in S_{k+1}$, define the number of common faces between $\sigma$ and $\partial_{k+1}\tau$ by:

$$w(\sigma, \partial_{k+1}\tau) = \langle (\partial_{k+1}\tau)^*, \sigma \rangle_{C_k, C_k}^+$$

where $x^+ = \max(x, 0)$ for $x \in \mathbb{R}$. For another chain $\sigma'$, we say that $\sigma$ and $\sigma'$ are adjacent (in the sense that the random walk can reach $\sigma'$ from the state $\sigma$) and write

$$\sigma \sim \sigma' \iff \exists \tau \in S_{k+1}, w(\sigma, \partial_{k+1}\tau) > 0 \text{ and } \sigma' = \sigma - \partial_{k+1}\tau. \quad (22)$$

Finally, let us define the weight of the transition from $\sigma$ to $\sigma'$:

$$K(\sigma, \sigma') = \begin{cases} 1 & \text{if } \sigma = \sigma' = 0 \\ w(\sigma, \sigma - \sigma') & \text{if } \sigma \sim \sigma' \\ 0 & \text{otherwise.} \end{cases} \quad (23)$$

**Example 14.** In Figure 3, we see how a difference of orientations is simply reflected in the value of the scalar product. Note also that $w(\sigma, \sigma - \sigma')$ can be viewed as a scalar product that counts the number of edges of the triangle which are adjacent to the chain with the good orientations.

<figure><img src="image_4.png" /><figcaption>Figure 3: Different cases of orientations: $\tau = [v_1v_2v_3]$. Here $\sigma$ is a 1-chain, not necessarily a cycle. (a) In this case, $\sigma = [v_1v_2] + [v_2v_3]$ and $w(\sigma, \partial_2\tau) = 2$, which is the number of edges in common between $\sigma$ and $\tau$. (b) Here, $\sigma = [v_1v_2] - [v_2v_3]$ and $w(\sigma, \partial_2\tau) = 0$. So $\tau$ is never chosen for defining the transition to the next step here. (c) In the case (a), the next step is $\sigma' = [v_1v_2] + [v_2v_3] - \partial_2\tau = [v_1v_3]$.</figcaption></figure>

**Generator of the random walk** Let us define by $(\mathcal{A}, D(\mathcal{A}))$ the generator of the continuous-time random walk.

**Definition 15** (Cycle-valued random walk). Let $D(\mathcal{A})$ be the set of functions $F$ such that $|\sum_{\sigma' \sim \sigma} (F(\sigma') - F(\sigma)) K(\sigma, \sigma')| < +\infty$. For $F \in D(\mathcal{A})$, we can define

$$AF: C_k \to \mathbb{R} \\ \sigma \mapsto \sum_{\sigma' \sim \sigma} (F(\sigma') - F(\sigma)) K(\sigma, \sigma').$$

Let us consider a Lipschitz continuous function $F$ from $C_k$ to $\mathbb{R}$, i.e. such that there exists $c_F > 0$ such that for any $\sigma, \sigma' \in C_k$, such that $\sigma \sim \sigma'$, we have

$$|F(\sigma) - F(\sigma')| \le c_F \|\sigma - \sigma'\|_{C_k} = c_F \|\partial_{k+1}\tau\|_{C_k} = c_F \sqrt{k+2},$$
**Transition kernel.** Let us explain the transition of our chain-valued random walk. For the sake of simplicity, imagine here that $k = 1$. Assume that we are in a state $\sigma \in C_1$ (a cycle for $k = 1$). Because transitions are in $\text{im } \partial_2$, let us consider an element of this space, say $\partial_2\tau$ for $\tau \in S_2$ (a triangle). $\partial_2\tau$ defines a possible transition if $\tau$ is upper-adjacent to $\sigma$, i.e. if $\tau$ and $\sigma$ share at least one edge. All the possible transitions from $\sigma$ are obtained by letting $\tau$ vary in $S_2$. The more $\tau$ and $\sigma$ have edges in common and the more $\tau$ will be likely to define the next step of the random walk and we thus need to define a weight to account for this.

For a k-chain $\sigma \neq 0$ and an oriented $(k+1)$-simplex $\tau \in S_{k+1}$, define the number of common faces between $\sigma$ and $\partial_{k+1}\tau$ by:

$$w(\sigma, \partial_{k+1}\tau) = \langle (\partial_{k+1}\tau)^*, \sigma \rangle_{C_k, C_k}^+$$

where $x^+ = \max(x, 0)$ for $x \in \mathbb{R}$. For another chain $\sigma'$, we say that $\sigma$ and $\sigma'$ are adjacent (in the sense that the random walk can reach $\sigma'$ from the state $\sigma$) and write

$$\sigma \sim \sigma' \iff \exists \tau \in S_{k+1}, w(\sigma, \partial_{k+1}\tau) > 0 \text{ and } \sigma' = \sigma - \partial_{k+1}\tau. \quad (22)$$

Finally, let us define the weight of the transition from $\sigma$ to $\sigma'$:

$$K(\sigma, \sigma') = \begin{cases} 1 & \text{if } \sigma = \sigma' = 0 \\ w(\sigma, \sigma - \sigma') & \text{if } \sigma \sim \sigma' \\ 0 & \text{otherwise.} \end{cases} \quad (23)$$

**Example 14.** In Figure 3, we see how a difference of orientations is simply reflected in the value of the scalar product. Note also that $w(\sigma, \sigma - \sigma')$ can be viewed as a scalar product that counts the number of edges of the triangle which are adjacent to the chain with the good orientations.

<figure><img src="image_6.png" /><figcaption>Figure 3: Different cases of orientations: $\tau = [v_1v_2v_3]$. Here $\sigma$ is a 1-chain, not necessarily a cycle. (a) In this case, $\sigma = [v_1v_2] + [v_2v_3]$ and $w(\sigma, \partial_2\tau) = 2$, which is the number of edges in common between $\sigma$ and $\tau$. (b) Here, $\sigma = [v_1v_2] - [v_2v_3]$ and $w(\sigma, \partial_2\tau) = 0$. So $\tau$ is never chosen for defining the transition to the next step here. (c) In the case (a), the next step is $\sigma' = [v_1v_2] + [v_2v_3] - \partial_2\tau = [v_1v_3]$.</figcaption></figure>

**Generator of the random walk** Let us define by $(\mathcal{A}, D(\mathcal{A}))$ the generator of the continuous-time random walk.

**Definition 15** (Cycle-valued random walk). Let $D(\mathcal{A})$ be the set of functions $F$ such that $|\sum_{\sigma' \sim \sigma} (F(\sigma') - F(\sigma)) K(\sigma, \sigma')| < +\infty$. For $F \in D(\mathcal{A})$, we can define

$$AF: C_k \to \mathbb{R} \\ \sigma \mapsto \sum_{\sigma' \sim \sigma} (F(\sigma') - F(\sigma)) K(\sigma, \sigma').$$

Let us consider a Lipschitz continuous function $F$ from $C_k$ to $\mathbb{R}$, i.e. such that there exists $c_F > 0$ such that for any $\sigma, \sigma' \in C_k$, such that $\sigma \sim \sigma'$, we have

$$|F(\sigma) - F(\sigma')| \le c_F \|\sigma - \sigma'\|_{C_k} = c_F \|\partial_{k+1}\tau\|_{C_k} = c_F \sqrt{k+2},$$
because the boundary of $\tau \in C_{k+1}$ contains $k+2-k$ simplexes. Moreover, we remark that if $\sigma$ and $\sigma' \in C_k$,

$$ \sigma \sim \sigma' \implies K(\sigma, \sigma') \le \sqrt{k+1} \|\sigma\|_{C_k}. $$

Indeed, $\sigma' - \sigma$ has $k+1$ faces of weight 1, so $K(\sigma, \sigma')$ is bounded by the sum of the weights of the faces of $\sigma$. Hence,

$$ \left| \sum_{\sigma' \sim \sigma} (F(\sigma') - F(\sigma)) K(\sigma, \sigma') \right| \le c_F(k+2) \|\sigma\|_{C_k}, $$

from which we deduce that $F \in D(\mathcal{A})$.

It is immediate that (see Ethier and Kurtz [17, Chapter 4, Section 2 and Chapter 8, Section 3]):

**Theorem 16.** The map $\mathcal{A}$ of domain $D(\mathcal{A})$ generates a strong Feller continuous Markov process $X = (X_t)_{t \ge 0}$ on $C_b(C_k, \mathbf{R})$, the set of continuous bounded functions on $C_k$. The set $\mathcal{A} = \mathcal{D} \cap C_b(C_k, \mathbf{R})$ is a core for $X$.

Remark that the process $X$ is a continuous-time pure jump process and admits a representation with a discrete-time Markov chain and exponentially distributed clocks attached with each possible transitions (see e.g. [17, Chapter 4, Section 2]).

**Theorem 17.** For any $t > 0$, $X_t$ remains in the same homology class as $X_0$. Moreover, if $X_0 \in \ker \partial_k$, then for any $t \ge 0$, $X_t$ belongs to $\ker \partial_k$.

*Proof.* At each change of state, we add to $X$ an element of $\mathrm{im} \, \partial_{k+1}$. Since $\partial_k \circ \partial_{k+1} = 0$, we add only elements of $\ker \partial_k$ to $X_0$, hence $X_t$ always belongs to $\ker \partial_k$ and the homology class does not change along the dynamics of $X$. ■

We can precise the link between $\mathcal{A}$ and $L_k^\uparrow$. Note that they cannot be equal since $\mathcal{A}$ operate on functions of chains whereas $L_k^\uparrow$ operates on cochains $C^k$, i.e. linear functions of chains. However we will see that $\mathcal{A}$, $-L_k$ and $-L_k^\uparrow$ coincide when restricted to $\ker \partial_k$.

**Theorem 18.** We have for every $\zeta \in \ker \partial_k$, and for any $\sigma \in C_k$,

$$ \mathcal{A}\zeta(\sigma) = -L_k^\uparrow\zeta(\sigma) = -L_k\zeta(\sigma). \qquad (24) $$

*Proof.* From (15), we have:

$$ \mathcal{A}\zeta(\sigma) = \sum_{\sigma' \sim \sigma} (\zeta(\sigma') - \zeta(\sigma)) K(\sigma, \sigma'). \qquad (25) $$

By the definitions of the $\sim$ relation (22) and of $K$ (23):

$$ \begin{aligned} \sum_{\sigma' \sim \sigma} (\zeta(\sigma') - \zeta(\sigma)) K(\sigma, \sigma') &= - \langle \zeta, \sum_{\tau \in S_{k+1}} \langle (\partial_{k+1} \tau)^*, \sigma \rangle_{C^k, C_k}^+ \partial_{k+1} \tau \rangle_{C^k, C_k} \\ &= - \langle \partial_{k+1}^* \zeta, \sum_{\tau \in S_{k+1}} \langle (\partial_{k+1} \tau)^*, \sigma \rangle_{C^k, C_k}^+ \tau \rangle_{C^{k+1}, C_{k+1}} \\ &= - \langle \partial_{k+1}^* \zeta, \sum_{\tau \in S_{k+1}} \langle \tau^*, \partial_{k+1}^* \sigma \rangle_{C^k, C_k}^+ \tau \rangle_{C^{k+1}, C_{k+1}} \\ &= - \langle \partial_{k+1}^* \zeta, \partial_{k+1}^* \sigma \rangle_{C^{k+1}, C_{k+1}} \\ &= - \langle \partial_{k+1} \circ \partial_{k+1}^* \zeta, \sigma \rangle_{C^{k+1}, C_{k+1}} \\ &= - L_k^\uparrow \zeta(\sigma). \end{aligned} \qquad (26) $$

When $\sigma \in \ker \partial_k$, $L_k^\downarrow(\tau) = 0$. This concludes the proof. ■
because the boundary of $\tau \in C_{k+1}$ contains $k+2-k$ simplexes. Moreover, we remark that if $\sigma$ and $\sigma' \in C_k$,

$$ \sigma \sim \sigma' \implies K(\sigma, \sigma') \le \sqrt{k+1} \|\sigma\|_{C_k}. $$

Indeed, $\sigma' - \sigma$ has $k+1$ faces of weight 1, so $K(\sigma, \sigma')$ is bounded by the sum of the weights of the faces of $\sigma$. Hence,

$$ \left| \sum_{\sigma' \sim \sigma} (F(\sigma') - F(\sigma)) K(\sigma, \sigma') \right| \le c_F(k+2) \|\sigma\|_{C_k}, $$

from which we deduce that $F \in D(\mathcal{A})$.

It is immediate that (see Ethier and Kurtz [17, Chapter 4, Section 2 and Chapter 8, Section 3]):

**Theorem 16.** The map $\mathcal{A}$ of domain $D(\mathcal{A})$ generates a strong Feller continuous Markov process $X = (X_t)_{t \ge 0}$ on $C_b(C_k, \mathbf{R})$, the set of continuous bounded functions on $C_k$. The set $\mathcal{A} = \mathcal{D} \cap C_b(C_k, \mathbf{R})$ is a core for $X$.

Remark that the process $X$ is a continuous-time pure jump process and admits a representation with a discrete-time Markov chain and exponentially distributed clocks attached with each possible transitions (see e.g. [17, Chapter 4, Section 2]).

**Theorem 17.** For any $t > 0$, $X_t$ remains in the same homology class as $X_0$. Moreover, if $X_0 \in \ker \partial_k$, then for any $t \ge 0$, $X_t$ belongs to $\ker \partial_k$.

*Proof.* At each change of state, we add to $X$ an element of $\mathrm{im} \, \partial_{k+1}$. Since $\partial_k \circ \partial_{k+1} = 0$, we add only elements of $\ker \partial_k$ to $X_0$, hence $X_t$ always belongs to $\ker \partial_k$ and the homology class does not change along the dynamics of $X$. ■

We can precise the link between $\mathcal{A}$ and $L_k^\uparrow$. Note that they cannot be equal since $\mathcal{A}$ operate on functions of chains whereas $L_k^\uparrow$ operates on cochains $C^k$, i.e. linear functions of chains. However we will see that $\mathcal{A}$, $-L_k$ and $-L_k^\uparrow$ coincide when restricted to $\ker \partial_k$.

**Theorem 18.** We have for every $\zeta \in \ker \partial_k$, and for any $\sigma \in C_k$,

$$ \mathcal{A}\zeta(\sigma) = -L_k^\uparrow\zeta(\sigma) = -L_k\zeta(\sigma). \qquad (24) $$

*Proof.* From (15), we have:

$$ \mathcal{A}\zeta(\sigma) = \sum_{\sigma' \sim \sigma} (\zeta(\sigma') - \zeta(\sigma)) K(\sigma, \sigma'). \qquad (25) $$

By the definitions of the $\sim$ relation (22) and of $K$ (23):

$$ \begin{aligned} \sum_{\sigma' \sim \sigma} (\zeta(\sigma') - \zeta(\sigma)) K(\sigma, \sigma') &= - \langle \zeta, \sum_{\tau \in S_{k+1}} \langle (\partial_{k+1} \tau)^*, \sigma \rangle_{C^k, C_k}^+ \partial_{k+1} \tau \rangle_{C^k, C_k} \\ &= - \langle \partial_{k+1}^* \zeta, \sum_{\tau \in S_{k+1}} \langle (\partial_{k+1} \tau)^*, \sigma \rangle_{C^k, C_k}^+ \tau \rangle_{C^{k+1}, C_{k+1}} \\ &= - \langle \partial_{k+1}^* \zeta, \sum_{\tau \in S_{k+1}} \langle \tau^*, \partial_{k+1}^* \sigma \rangle_{C^k, C_k}^+ \tau \rangle_{C^{k+1}, C_{k+1}} \\ &= - \langle \partial_{k+1}^* \zeta, \partial_{k+1}^* \sigma \rangle_{C^{k+1}, C_{k+1}} \\ &= - \langle \partial_{k+1} \circ \partial_{k+1}^* \zeta, \sigma \rangle_{C^{k+1}, C_{k+1}} \\ &= - L_k^\uparrow \zeta(\sigma). \end{aligned} \qquad (26) $$

When $\sigma \in \ker \partial_k$, $L_k^\downarrow(\tau) = 0$. This concludes the proof. ■
<figure><img src="image_5.png" /><figcaption>Figure 4: Example illustrating the occurrence of edges with weights greater than 1 in the triangulation of the torus. Imagine a cycle that is composed of the blue-green curve, and then the red-orange curve. If the red edge is selected to determine the next step of the chain, and if the lower triangle adjacent to it is chosen, then the green edge will have a weight equal to two.</figcaption></figure>

## 3.2 Non-explosion and recurrence of the chain on finite simplicial complexes

In contrast with the 0-dimensional case, there is no reason why the state space of the stochastic process $X$ should be finite in the general case, even when $S_k$ is, since the weights $\lambda_\tau$ in (3) can be unbounded. For a cycle random walk, this means for instance, that the cycle can do loops and go several times through the same edge. Even in a simple structure such as the triangulation of the torus, this can happen.

Fortunately, the dynamics of the random walk tend to limit its length.

**Definition 19.** For a k-chain $\sigma$, we denote $\mathcal{R}(\sigma)$, the recurrence class of $\sigma$, consisting of all the k-chains which can be attained by $X$ starting from $\sigma$.

**Theorem 20.** Suppose $S_{k+1}$ is finite and $X_0$ is a k-cycle. Then, the process $(X_t)_{t \ge 0}$ on $\mathcal{R}(X_0)$ is non-explosive, recurrent and admits a unique invariant measure $\pi$. Furthermore,

$$ \int_{C_k} \|\sigma\|_{C_k}^2 d\pi(\sigma) < \infty. $$

*Proof.* To prove the result, we are going to show that $\|\cdot\|_{C_k}^2$ is a Lyapunov function for $\mathcal{A}$. Let $P_{\text{ker}}$, $P_{\text{im}}$ : $C_k \to C_k$ be the projection operators of k-chains to $\text{ker } L_k^\uparrow$ and $\text{im } L_k^\uparrow$ respectively. For $\sigma \in \mathcal{R}(X_0)$ we have

$$ \begin{aligned} (\mathcal{A} \| \cdot \|_{C_k}^2)(\sigma) &= \sum_{\tau \in S_{k+1}} (\|\sigma - \partial_{k+1}\tau\|_{C_k}^2 - \|\sigma\|_{C_k}^2) K(\sigma, \sigma - \tau) \\ &= \sum_{\tau \in S_{k+1}} (-2 \langle (\partial_{k+1}\tau)^*, \sigma \rangle_{C^k, C_k} + \|\partial_{k+1}\tau\|_{C_k}^2) K(\sigma, \sigma - \tau) \\ &= 2 \langle \mathcal{A}\sigma^*, \sigma \rangle_{C^k, C_k} + (k+2) \sum_{\tau \in S_{k+1}} \langle (\partial_{k+1}\tau)^*, \sigma \rangle_{C^k, C_k}^+ \\ &= -2 \langle L_k^\uparrow \sigma^*, \sigma \rangle_{C^k, C_k} + (k+2) \sum_{\tau \in S_{k+1}} \langle \tau^*, \partial_{k+1}^* \sigma \rangle_{C^k, C_k}^+ . \end{aligned} \quad (27) $$

The factor $k+2$ in the third line comes from the fact that $\|\partial_{k+1}\tau\|_{C_k}^2 = k+2$. For the fourth equality, we have used Theorem 18 from which:

$$ \langle \mathcal{A}\sigma^*, \sigma \rangle_{C^k, C_k} = - \langle L_k^\uparrow \sigma^*, \sigma \rangle_{C^k, C_k}. $$

Since $S_k$ is finite, $L_k^\uparrow$ is a finite dimensional operator with positive discrete eigenvalues. Thus, denoting by $\lambda_m$ the smallest positive eigenvalue of $L_k^\uparrow$, we have

$$ \langle L_k^\uparrow P_{\text{im}} \sigma^*, P_{\text{im}} \sigma \rangle_{C^k, C_k} \ge \lambda_m \|P_{\text{im}} \sigma\|_{C^k}^2. $$
<figure><img src="image_7.png" /><figcaption>Figure 4: Example illustrating the occurrence of edges with weights greater than 1 in the triangulation of the torus. Imagine a cycle that is composed of the blue-green curve, and then the red-orange curve. If the red edge is selected to determine the next step of the chain, and if the lower triangle adjacent to it is chosen, then the green edge will have a weight equal to two.</figcaption></figure>

## 3.2 Non-explosion and recurrence of the chain on finite simplicial complexes

In contrast with the 0-dimensional case, there is no reason why the state space of the stochastic process $X$ should be finite in the general case, even when $S_k$ is, since the weights $\lambda_\tau$ in (3) can be unbounded. For a cycle random walk, this means for instance, that the cycle can do loops and go several times through the same edge. Even in a simple structure such as the triangulation of the torus, this can happen.

Fortunately, the dynamics of the random walk tend to limit its length.

**Definition 19.** For a k-chain $\sigma$, we denote $\mathcal{R}(\sigma)$, the recurrence class of $\sigma$, consisting of all the k-chains which can be attained by $X$ starting from $\sigma$.

**Theorem 20.** Suppose $S_{k+1}$ is finite and $X_0$ is a k-cycle. Then, the process $(X_t)_{t \ge 0}$ on $\mathcal{R}(X_0)$ is non-explosive, recurrent and admits a unique invariant measure $\pi$. Furthermore,

$$ \int_{C_k} \|\sigma\|_{C_k}^2 d\pi(\sigma) < \infty. $$

*Proof.* To prove the result, we are going to show that $\|\cdot\|_{C_k}^2$ is a Lyapunov function for $\mathcal{A}$. Let $P_{\text{ker}}$, $P_{\text{im}}$ : $C_k \to C_k$ be the projection operators of k-chains to $\text{ker } L_k^\uparrow$ and $\text{im } L_k^\uparrow$ respectively. For $\sigma \in \mathcal{R}(X_0)$ we have

$$ \begin{aligned} (\mathcal{A} \| \cdot \|_{C_k}^2)(\sigma) &= \sum_{\tau \in S_{k+1}} (\|\sigma - \partial_{k+1}\tau\|_{C_k}^2 - \|\sigma\|_{C_k}^2) K(\sigma, \sigma - \tau) \\ &= \sum_{\tau \in S_{k+1}} (-2 \langle (\partial_{k+1}\tau)^*, \sigma \rangle_{C^k, C_k} + \|\partial_{k+1}\tau\|_{C_k}^2) K(\sigma, \sigma - \tau) \\ &= 2 \langle \mathcal{A}\sigma^*, \sigma \rangle_{C^k, C_k} + (k+2) \sum_{\tau \in S_{k+1}} \langle (\partial_{k+1}\tau)^*, \sigma \rangle_{C^k, C_k}^+ \\ &= -2 \langle L_k^\uparrow \sigma^*, \sigma \rangle_{C^k, C_k} + (k+2) \sum_{\tau \in S_{k+1}} \langle \tau^*, \partial_{k+1}^* \sigma \rangle_{C^k, C_k}^+ . \end{aligned} \quad (27) $$

The factor $k+2$ in the third line comes from the fact that $\|\partial_{k+1}\tau\|_{C_k}^2 = k+2$. For the fourth equality, we have used Theorem 18 from which:

$$ \langle \mathcal{A}\sigma^*, \sigma \rangle_{C^k, C_k} = - \langle L_k^\uparrow \sigma^*, \sigma \rangle_{C^k, C_k}. $$

Since $S_k$ is finite, $L_k^\uparrow$ is a finite dimensional operator with positive discrete eigenvalues. Thus, denoting by $\lambda_m$ the smallest positive eigenvalue of $L_k^\uparrow$, we have

$$ \langle L_k^\uparrow P_{\text{im}} \sigma^*, P_{\text{im}} \sigma \rangle_{C^k, C_k} \ge \lambda_m \|P_{\text{im}} \sigma\|_{C^k}^2. $$
On the other hand, by Cauchy-Schwarz inequality:

$$ \sum_{\tau \in S_{k+1}} \langle (\partial_{k+1} \tau)^*, \sigma \rangle_{C^k, C_k}^+ = \sum_{\tau \in S_{k+1}} \langle \tau^*, \partial_{k+1}^* \sigma \rangle_{C^{k+1}, C_{k+1}}^+ \\ \le |S_{k+1}|^{1/2} \left( \sum_{\tau \in S_{k+1}} \langle \tau^*, \partial_{k+1}^* \sigma \rangle_{C^{k+1}, C_{k+1}}^2 \right)^{1/2} = |S_{k+1}|^{1/2} \|\partial_{k+1}^* \sigma\|_{C^{k+1}}. \quad (28) $$

Then:

$$ \| \partial_{k+1}^* \sigma \|_{C^{k+1}}^2 = \langle \partial_{k+1}^* \sigma^*, \partial_{k+1}^* \sigma^* \rangle_{C^{k+1}, C^{k+1}} = \langle L_k^\uparrow \sigma^*, \sigma^* \rangle_{C^k, C^k} \le \lambda_M \|\sigma\|_{C_k}^2, \quad (29) $$

where $\lambda_M$ is the largest eigenvalue of $L_k^\uparrow$. (28) and (29) imply that:

$$ \sum_{\tau \in S_{k+1}} \langle (\partial_{k+1} \tau)^*, \sigma \rangle_{C^k, C_k}^+ \le \sqrt{\lambda_M |S_{k+1}|} \|\sigma\|_{C_k}. \quad (30) $$

Now, since the transitions of $X$ are in $im L_k^\uparrow$, we have that

$$ \forall \sigma \in \mathcal{R}(X_0), \quad \sigma = P_{\ker} X_0 + P_{\text{im}} \sigma $$

thus

$$ \| P_{\text{im}} \sigma \|_{C_k}^2 = \| \sigma \|_{C_k}^2 - \| P_{\ker} X_0 \|_{C_k}^2. $$

Combining everything together, we have

$$ \begin{aligned} (\mathcal{A} \|\cdot\|_{C_k}^2)(\sigma) & \le -2 \langle L_k^\uparrow P_{\text{im}} \sigma^*, P_{\text{im}} \sigma \rangle_{C^k, C_k} + (k+2) |S_{k+1}|^{1/2} \lambda_M^{1/2} \|\sigma\|_{C_k} \\ & \le -2\lambda_m \|P_{\text{im}} \sigma\|_{C_k}^2 + (k+2) |S_{k+1}|^{1/2} \lambda_M^{1/2} \|\sigma\|_{C_k} \\ & \le -2\lambda_m \|\sigma\|_{C_k}^2 + (k+2) |S_{k+1}|^{1/2} \lambda_M^{1/2} \|\sigma\|_{C_k} + 2\lambda_m \|P_{\ker} X_0\|_{C_k}^2 \\ & \le -\lambda_m \|\sigma\|_{C_k}^2 + ((k+2) |S_{k+1}|^{1/2} \lambda_M^{1/2} \|\sigma\|_{C_k} - \lambda_m \|\sigma\|_{C_k}^2) + 2\lambda_m \|P_{\ker} X_0\|_{C_k}^2 \\ & \le -\lambda_m \|\sigma\|_{C_k}^2 + \frac{(k+2)^2 |S_{k+1}| \lambda_M}{4\lambda_m} + 2\lambda_m \|P_{\ker} X_0\|_{C_k}^2, \end{aligned} \quad (31) $$

where the last inequality comes from the fact that the function

$$ x \mapsto (k+2) |S_{k+1}|^{1/2} \lambda_M^{1/2} x - \lambda_m x^2 $$

reaches its maximum at $x = (k+2)|S_{k+1}|^{1/2}\lambda_M^{1/2}/(2\lambda_m)$ at which point the value of the function is $(k+2)^2|S_{k+1}|\lambda_M/(4\lambda_m)$. Thus, $\|\cdot\|_{C_k}^2$ is a Lyapunov function for $\mathcal{A}$ and the results then follow by applying Theorems 2.1, 4.2 and 4.3 from Meyn and Tweedie [37]. ■

The computation (31) in the proof of Theorem 20 can be further exploited to show that the expectation of the square norm remains bounded along the chain:

**Corollary 21.** Suppose that $S_{k+1}$ and $\mathbb{E}(\|X_0\|_{C_k}^2)$ are finite. Then, for $T > 0$, and for any $t \in [0, T]$,

$$ \mathbb{E}(\|X_t\|_{C_k}^2) \le \left[ (1 + 2\lambda_m T)\mathbb{E}(\|X_0\|_{C_k}^2) + T \frac{(k+2)^2 |S_k| \lambda_M}{4\lambda_m} \right] e^{-\lambda_m t}. \quad (32) $$

*Proof.* For $M > 0$, let us consider the stopping time $S_M = \inf\{\|X_t\|_{C_k}^2 > M\}$. Using Itô's formula [26, Th.5.1 P.66]:

$$ \begin{aligned} \mathbb{E}(\|X_{t \wedge S_M}\|_{C_k}^2) &= \mathbb{E}(\|X_0\|_{C_k}^2) + \mathbb{E}\left(\int_0^{t \wedge S_M} (\mathcal{A} \|\cdot\|_{C_k}^2)(X_s) ds\right) \\ &\le \mathbb{E}(\|X_0\|_{C_k}^2) - \lambda_m \int_0^t \mathbb{E}(\|X_{s \wedge S_M}\|_{C_k}^2) ds \\ &\quad + T \left[ \frac{(k+2)^2 |S_{k+1}| \lambda_M}{4\lambda_m} + 2\lambda_m \|P_{\ker} X_0\|_{C_k}^2 \right] \\ &\le \left[ \mathbb{E}(\|X_0\|_{C_k}^2) + T \frac{(k+2)^2 |S_{k+1}| \lambda_M}{4\lambda_m} + 2\lambda_m T \mathbb{E}(\|P_{\ker} X_0\|_{C_k}^2) \right] e^{-\lambda_m t}, \end{aligned} $$
On the other hand, by Cauchy-Schwarz inequality:

$$
\sum_{\tau \in S_{k+1}} \langle (\partial_{k+1} \tau)^*, \sigma \rangle_{C^k, C_k}^+ = \sum_{\tau \in S_{k+1}} \langle \tau^*, \partial_{k+1}^* \sigma \rangle_{C^{k+1}, C_{k+1}}^+ \\
\le |S_{k+1}|^{1/2} \left( \sum_{\tau \in S_{k+1}} \langle \tau^*, \partial_{k+1}^* \sigma \rangle_{C^{k+1}, C_{k+1}}^2 \right)^{1/2} = |S_{k+1}|^{1/2} \|\partial_{k+1}^* \sigma\|_{C^{k+1}}. \quad (28)
$$

Then:

$$
\| \partial_{k+1}^* \sigma \|_{C^{k+1}}^2 = \langle \partial_{k+1}^* \sigma^*, \partial_{k+1}^* \sigma^* \rangle_{C^{k+1}, C^{k+1}} = \langle L_k^\uparrow \sigma^*, \sigma^* \rangle_{C^k, C^k} \le \lambda_M \|\sigma\|_{C_k}^2, \quad (29)
$$

where $\lambda_M$ is the largest eigenvalue of $L_k^\uparrow$. (28) and (29) imply that:

$$
\sum_{\tau \in S_{k+1}} \langle (\partial_{k+1} \tau)^*, \sigma \rangle_{C^k, C_k}^+ \le \sqrt{\lambda_M |S_{k+1}|} \|\sigma\|_{C_k}. \quad (30)
$$

Now, since the transitions of X are in im $L_k^{\uparrow}$, we have that

$$
\forall \sigma \in \mathcal{R}(X_0), \quad \sigma = P_{\ker}X_0 + P_{\mathrm{im}}\sigma
$$

thus

$$
\| P_{\mathrm{im}} \sigma \|_{C_k}^2 = \| \sigma \|_{C_k}^2 - \| P_{\ker} X_0 \|_{C_k}^2.
$$

Combining everything together, we have

$$
\begin{align*}
(\mathcal{A}\|\cdot\|_{C_k}^2)(\sigma) &\le -2 \langle L_k^\uparrow P_{\mathrm{im}}\sigma^*, P_{\mathrm{im}}\sigma \rangle_{C^k, C_k} + (k+2)|S_{k+1}|^{1/2} \lambda_M^{1/2} \|\sigma\|_{C_k} \\
&\le -2\lambda_m \|P_{\mathrm{im}}\sigma\|_{C_k}^2 + (k+2)|S_{k+1}|^{1/2} \lambda_M^{1/2} \|\sigma\|_{C_k} \\
&\le -2\lambda_m \|\sigma\|_{C_k}^2 + (k+2)|S_{k+1}|^{1/2} \lambda_M^{1/2} \|\sigma\|_{C_k} + 2\lambda_m \|P_{\ker}X_0\|_{C_k}^2 \\
&\le -\lambda_m \|\sigma\|_{C_k}^2 + ((k+2)|S_{k+1}|^{1/2} \lambda_M^{1/2} \|\sigma\|_{C_k} - \lambda_m \|\sigma\|_{C_k}^2) + 2\lambda_m \|P_{\ker}X_0\|_{C_k}^2 \\
&\le -\lambda_m \|\sigma\|_{C_k}^2 + \frac{(k+2)^2|S_{k+1}|\lambda_M}{4\lambda_m} + 2\lambda_m \|P_{\ker}X_0\|_{C_k}^2, \tag{31}
\end{align*}
$$

where the last inequality comes from the fact that the function

$$
x \mapsto (k+2) |S_{k+1}|^{1/2} \lambda_M^{1/2} x - \lambda_m x^2
$$

reaches its maximum at $x = (k+2)|S_{k+1}|^{1/2}\lambda_M^{1/2}/(2\lambda_m)$ at which point the value of the function is $(k+2)^2|S_{k+1}|\lambda_M/(4\lambda_m)$. Thus, $\|\cdot\|_{C_k}^2$ is a Lyapunov function for $\mathcal{A}$ and the results then follow by applying Theorems 2.1, 4.2 and 4.3 from Meyn and Tweedie [37]. ■

The computation (31) in the proof of Theorem 20 can be further exploited to show that the expectation of the square norm remains bounded along the chain:

**Corollary 21.** Suppose that $S_{k+1}$ and $\mathbb{E}(\|X_0\|_{C_k}^2)$ are finite. Then, for $T > 0$, and for any $t \in [0, T]$,

$$
\mathbb{E}(\|X_t\|_{C_k}^2) \le \left[ (1 + 2\lambda_m T)\mathbb{E}(\|X_0\|_{C_k}^2) + T \frac{(k+2)^2 |S_k| \lambda_M}{4\lambda_m} \right] e^{-\lambda_m t}. \quad (32)
$$

*Proof.* For $M > 0$, let us consider the stopping time $S_M = \inf\{\|X_t\|_{C_k}^2 > M\}$. Using Itô's formula [26, Th.5.1 P.66]:

$$
\begin{align*}
\mathbb{E}(\|X_{t \wedge S_M}\|_{C_k}^2) &= \mathbb{E}(\|X_0\|_{C_k}^2) + \mathbb{E}\left(\int_0^{t \wedge S_M} (\mathcal{A}\|\cdot\|_{C_k}^2)(X_s) ds\right) \\
&\le \mathbb{E}(\|X_0\|_{C_k}^2) - \lambda_m \int_0^t \mathbb{E}(\|X_{s \wedge S_M}\|_{C_k}^2) ds \\
&\quad + T \left[ \frac{(k+2)^2 |S_{k+1}| \lambda_M}{4\lambda_m} + 2\lambda_m \|P_{\ker}X_0\|_{C_k}^2 \right] \\
&\le \left[ \mathbb{E}(\|X_0\|_{C_k}^2) + T \frac{(k+2)^2 |S_{k+1}| \lambda_M}{4\lambda_m} + 2\lambda_m T \mathbb{E}(\|P_{\ker}X_0\|_{C_k}^2) \right] e^{-\lambda_m t},
\end{align*}
$$
by Gronwall's lemma. Because the right hand side does not depend on $M$, we deduce that $\lim_{M \to +\infty} S_M = +\infty$ almost surely and the result follows. ■

We can go a little further. For any $k + 1$-simplex $\tau$, let $\deg_{\downarrow}(\tau)$ be the lower adjacency degree of $\tau$ given by

$$ \deg_{\downarrow}(\tau) = \sum_{\tau' \in S_{k+1}, \tau' \neq \tau} |\langle (\partial_{k+1}\tau)^*, \partial_{k+1}\tau' \rangle_{C^k, C_k}|. \quad (33) $$

The quantity in the sum corresponds to the number of faces that $\tau$ and $\tau'$ have in common, so $\deg_{\downarrow}(\tau)$ counts the number of $k$-simplexes (with multiplicity) that share a face with $\tau$.

**Theorem 22.** Let $k \in \mathbb{N}$. Let $\sigma_1, \dots, \sigma_{\beta_k} \in C_k$ be a basis of $H_k$ and let $\tau_1, \dots, \tau_n \in S_{k+1}^+$ be the $k+1$-simplexes of our simplicial complex. Suppose that there exists $\sigma \in C_k$ and $(\lambda_{\tau}, \tau \in S_{k+1}^+) \in \{-1, 0, 1\}^{S_{k+1}^+}$ such that such that

$$ X_0 = \sigma + \sum_{\tau \in S_{k+1}^+} \lambda_{\tau} \partial_{k+1} \tau. $$

If we have for all $\tau \in S_{k+1}^+$ that:

$$ \deg_{\downarrow}(\tau) \leq k + 2 - |\langle (\partial_{k+1}\tau)^*, \sigma \rangle_{C^k, C_k}|, \quad (34) $$

then $X$ has a finite state space and for any $t \ge 0$, there exists $(\lambda_{\tau}(t), \tau \in S_{k+1}^+) \in \{-1, 0, 1\}^{S_{k+1}^+}$ such that

$$ X_t = \sigma + \sum_{\tau \in S_{k+1}^+} \lambda_{\tau}(t) \partial_{k+1} \tau. \quad (35) $$

This theorem ensures that under condition (34), the state space is finite and the chain does not 'loop' on itself too much as in the example of Fig. 4. This has direct consequences to determine its recurrence or to bound its length. In particular, if our simplicial complex is a triangulation of $\mathbb{R}^d$ with some holes, then (34) is verified for any chain $\sigma$ surrounding one (or several) holes.

**Proof of Theorem 22.** By the very definition of $X$, for $T_1$ the first jump time of $X$, there exist $(\lambda_{\tau}(T_1), \tau \in S_{k+1}^+)$ such that

$$ X_{T_1} = \sigma + \sum_{\tau \in S_{k+1}^+} \lambda_{\tau}(T_1) \partial_{k+1} \tau. $$

and $|\lambda_{\tau}(T_1) - \lambda_{\tau}| \le 1$ for any $\tau \in S_{k+1}^+$. First, consider $\eta \in S_{k+1}^+$ such that $\lambda_{\eta} = 1$. We have

$$ \begin{aligned} \langle (\partial_{k+1}\eta)^*, X_0 \rangle_{C^k, C_k} &= \langle (\partial_{k+1}\eta)^*, \sigma \rangle_{C^k, C_k} + \sum_{\tau \in S_{k+1}^+} \lambda_{\tau} \langle (\partial_{k+1}\eta)^*, \partial_{k+1}\tau \rangle_{C^k, C_k} \\ &\ge -|\langle (\partial_{k+1}\eta)^*, \sigma \rangle_{C^k, C_k}| + (k+2) - \sum_{\tau \neq \eta} |\langle (\partial_{k+1}\eta)^*, \partial_{k+1}\tau \rangle_{C^k, C_k}| \ge 0, \end{aligned} $$

by our assumption (34) and by the definition of $\deg_{\downarrow}(\tau)$ in (33). Thus,

$$ K(X_0, X_0 + \partial_{k+1}\eta) = w(X_0, -\partial_{k+1}\eta) = \max(0, -\langle (\partial_{k+1}\eta)^*, X_0 \rangle_{C^k, C_k}) = 0. \quad (36) $$

Hence, $\lambda_{\eta}(T_1)$ that was equal to 1 before the jump event can not increase and must be equal to 0 or to 1 after the jump.

Proceeding similarly for the simplexes $\eta \in S_{k+1}^+$ for which $\lambda_{\eta} = -1$, we obtain that $\lambda_{\tau}(T_1) \in \{-1, 0, 1\}$ for any $\tau \in S_{k+1}^+$. The proof is then concluded by induction with respect to the steps of the embedded discrete time Markov chain. ■

**Remark 23.** If the chain $X$ on a finite simplicial complex starts from a state $\tau \in \text{im} \, \partial_{k+1}$ then it is absorbed by the null chain in finite time.
by Gronwall's lemma. Because the right hand side does not depend on $M$, we deduce that $\lim_{M \to +\infty} S_M = +\infty$ almost surely and the result follows. ■

We can go a little further. For any $k + 1$-simplex $\tau$, let $\deg_{\downarrow}(\tau)$ be the lower adjacency degree of $\tau$ given by

$$ \deg_{\downarrow}(\tau) = \sum_{\tau' \in S_{k+1}, \tau' \neq \tau} |\langle (\partial_{k+1}\tau)^*, \partial_{k+1}\tau' \rangle_{C^k, C_k}|. \quad (33) $$

The quantity in the sum corresponds to the number of faces that $\tau$ and $\tau'$ have in common, so $\deg_{\downarrow}(\tau)$ counts the number of $k$-simplexes (with multiplicity) that share a face with $\tau$.

**Theorem 22.** Let $k \in \mathbb{N}$. Let $\sigma_1, \dots, \sigma_{\beta_k} \in C_k$ be a basis of $H_k$ and let $\tau_1, \dots, \tau_n \in S_{k+1}^+$ be the $k+1$-simplexes of our simplicial complex. Suppose that there exists $\sigma \in C_k$ and $(\lambda_{\tau}, \tau \in S_{k+1}^+) \in \{-1, 0, 1\}^{S_{k+1}^+}$ such that such that

$$ X_0 = \sigma + \sum_{\tau \in S_{k+1}^+} \lambda_{\tau} \partial_{k+1} \tau. $$

If we have for all $\tau \in S_{k+1}^+$ that:

$$ \deg_{\downarrow}(\tau) \leq k + 2 - |\langle (\partial_{k+1}\tau)^*, \sigma \rangle_{C^k, C_k}|, \quad (34) $$

then $X$ has a finite state space and for any $t \ge 0$, there exists $(\lambda_{\tau}(t), \tau \in S_{k+1}^+) \in \{-1, 0, 1\}^{S_{k+1}^+}$ such that

$$ X_t = \sigma + \sum_{\tau \in S_{k+1}^+} \lambda_{\tau}(t) \partial_{k+1} \tau. \quad (35) $$

This theorem ensures that under condition (34), the state space is finite and the chain does not 'loop' on itself too much as in the example of Fig. 4. This has direct consequences to determine its recurrence or to bound its length. In particular, if our simplicial complex is a triangulation of $\mathbb{R}^d$ with some holes, then (34) is verified for any chain $\sigma$ surrounding one (or several) holes.

**Proof of Theorem 22.** By the very definition of $X$, for $T_1$ the first jump time of $X$, there exist $(\lambda_{\tau}(T_1), \tau \in S_{k+1}^+)$ such that

$$ X_{T_1} = \sigma + \sum_{\tau \in S_{k+1}^+} \lambda_{\tau}(T_1) \partial_{k+1} \tau. $$

and $|\lambda_{\tau}(T_1) - \lambda_{\tau}| \le 1$ for any $\tau \in S_{k+1}^+$. First, consider $\eta \in S_{k+1}^+$ such that $\lambda_{\eta} = 1$. We have

$$ \begin{aligned} \langle (\partial_{k+1}\eta)^*, X_0 \rangle_{C^k, C_k} &= \langle (\partial_{k+1}\eta)^*, \sigma \rangle_{C^k, C_k} + \sum_{\tau \in S_{k+1}^+} \lambda_{\tau} \langle (\partial_{k+1}\eta)^*, \partial_{k+1}\tau \rangle_{C^k, C_k} \\ &\ge -|\langle (\partial_{k+1}\eta)^*, \sigma \rangle_{C^k, C_k}| + (k+2) - \sum_{\tau \neq \eta} |\langle (\partial_{k+1}\eta)^*, \partial_{k+1}\tau \rangle_{C^k, C_k}| \ge 0, \end{aligned} $$

by our assumption (34) and by the definition of $\deg_{\downarrow}(\tau)$ in (33). Thus,

$$ K(X_0, X_0 + \partial_{k+1}\eta) = w(X_0, -\partial_{k+1}\eta) = \max(0, -\langle (\partial_{k+1}\eta)^*, X_0 \rangle_{C^k, C_k}) = 0. \quad (36) $$

Hence, $\lambda_{\eta}(T_1)$ that was equal to 1 before the jump event can not increase and must be equal to 0 or to 1 after the jump.

Proceeding similarly for the simplexes $\eta \in S_{k+1}^+$ for which $\lambda_{\eta} = -1$, we obtain that $\lambda_{\tau}(T_1) \in \{-1, 0, 1\}$ for any $\tau \in S_{k+1}^+$. The proof is then concluded by induction with respect to the steps of the embedded discrete time Markov chain. ■

**Remark 23.** If the chain $X$ on a finite simplicial complex starts from a state $\tau \in \text{im} \partial_{k+1}$ then it is absorbed by the null chain in finite time.
It is well known that for random walks on undirected graphs, the stationary distribution gives to each vertex a weight proportional to its degree, i.e. to the number of edges it is adjacent to. In a way, the stationary measure highlights central nodes with respect to the 0-dimensional topology structure (i.e. connectivity). While we are not able to directly compute the stationary measure of our random walk, we would expect it to also highlight the topology of the simplicial complex. In Section 5, we will see that, empirically, the stationary measure of our random walk on edges tends highlight homology structures (in our case holes) by giving much weight to edges bordering the holes present in the homology class of our starting point. In practice our random walk would thus favor small chains circling the homology structure present in its homology class.

**Example 24.** To get some insights on how the situation can be complex when the state space is infinite, let us have a look at the random walk on the triangulation of the plane from which we have removed one triangle, so that the chain never dies. After a trillion of iterations, we get a graph similar to that of Figure 5 (a).

<figure><img src="image_6.png" /><figcaption>Figure 5: (a) A realization of $X$ after 2 trillions steps. The removed triangle is in red (center of the image). (b) The number of times $X$ touches the removed triangle, by packets of ten thousands steps. Simulation by M. Glisse.</figcaption></figure>

The support of the process $X$ is composed of several disconnected components, each of them may contain some holes. The isolated components are going to either die or merge with the component which contains the removed triangle. Provided that this is meaningful, if we look at the number of triangles which are inside the chain, it can increase or decrease by 1 with equal probability at each step. This means that it follows the law of a symmetric random walk, which is thus null recurrent. However, when the chain touches the boundary of the removed triangle, there is a drift only in the positive sense which ruins this reasoning. The simulation represented on Figure 5 (b) shows that $X$ touches the removed triangle very often.

# 4 Convergence

When dealing with random walks, it is natural to investigate their continuous diffusive limits. Here, we choose to embed our graph into another space and consider geometric random graphs (as is done e.g. in [59]). Limits of random walks on graphs drawn on a manifold have been considered for instance in [49, 21, 23, 25, 47] or [48], but the literature deals only with the convergence of generators as the latter is the key for applications in machine learning. In this section, we also study the tightness of the distributions of the random walks and show that the limiting values of the correctly rescaled cycle random walk are the diffusion solutions to a same martingale problem associated with the limit of the combinatorial Laplacian. As we will see in the computation done in this section, things can become quickly intricate,
It is well known that for random walks on undirected graphs, the stationary distribution gives to each vertex a weight proportional to its degree, i.e. to the number of edges it is adjacent to. In a way, the stationary measure highlights central nodes with respect to the 0-dimensional topology structure (i.e. connectivity). While we are not able to directly compute the stationary measure of our random walk, we would expect it to also highlight the topology of the simplicial complex. In Section 5, we will see that, empirically, the stationary measure of our random walk on edges tends highlight homology structures (in our case holes) by giving much weight to edges bordering the holes present in the homology class of our starting point. In practice our random walk would thus favor small chains circling the homology structure present in its homology class.

**Example 24.** To get some insights on how the situation can be complex when the state space is infinite, let us have a look at the random walk on the triangulation of the plane from which we have removed one triangle, so that the chain never dies. After a trillion of iterations, we get a graph similar to that of Figure 5 (a).

<figure><img src="image_8.png" /><figcaption>Figure 5: (a) A realization of $X$ after 2 trillions steps. The removed triangle is in red (center of the image). (b) The number of times $X$ touches the removed triangle, by packets of ten thousands steps. Simulation by M. Glisse.</figcaption></figure>

The support of the process $X$ is composed of several disconnected components, each of them may contain some holes. The isolated components are going to either die or merge with the component which contains the removed triangle. Provided that this is meaningful, if we look at the number of triangles which are inside the chain, it can increase or decrease by 1 with equal probability at each step. This means that it follows the law of a symmetric random walk, which is thus null recurrent. However, when the chain touches the boundary of the removed triangle, there is a drift only in the positive sense which ruins this reasoning. The simulation represented on Figure 5 (b) shows that $X$ touches the removed triangle very often.

# 4 Convergence

When dealing with random walks, it is natural to investigate their continuous diffusive limits. Here, we choose to embed our graph into another space and consider geometric random graphs (as is done e.g. in [59]). Limits of random walks on graphs drawn on a manifold have been considered for instance in [49, 21, 23, 25, 47] or [48], but the literature deals only with the convergence of generators as the latter is the key for applications in machine learning. In this section, we also study the tightness of the distributions of the random walks and show that the limiting values of the correctly rescaled cycle random walk are the diffusion solutions to a same martingale problem associated with the limit of the combinatorial Laplacian. As we will see in the computation done in this section, things can become quickly intricate,
and this is why we focus here on a particular case: we consider the scaling limits of the cycle random walk on the triangulation of the flat torus.

We denote by $\mathbb{T}_2$ the flat torus, which we embed into $\mathbb{R}^2$ as the rectangle $[0, 2] \times [0, \sqrt{3}]$ where the opposite edges are identified. Let $\epsilon_n = 1/n$ and consider

$$V_n = \{(2k\epsilon_n, 2l\sqrt{3}\epsilon_n), 0 \le k < n, 0 \le 2l < n\} \cup \{((2k+1)\epsilon_n, (2l+1)\sqrt{3}\epsilon_n), 0 < 2k+1 \le 2n, 0 < 2l+1 \le n\},$$

the set of vertices of the regular triangulation of mesh $2\epsilon_n$ and denote by $\mathcal{T}_n$ the triangulation based on $V_n$, see Figure 6. We consider the simplicial complex $\mathbb{C}^n$ composed of all the triangles of $\mathcal{T}_n$, their edges and their vertices $V_n$.

<figure><img src="image_7.png" /><figcaption>Figure 6: Regular triangulation of the flat torus. $0 := (0,0)$, $1 := (2\epsilon_n, 0)$, $2 := (\epsilon_n, \sqrt{3}\epsilon_n)$, $3 := (\epsilon_n, -\sqrt{3}\epsilon_n)$, $4 := (-\epsilon_n, \sqrt{3}\epsilon_n)$</figcaption></figure>

We will study here the cycle random walk $(X_t^n)_{t \ge 0}$ (correctly renormalized, as will be seen in the sequel) on this simplicial complex $\mathbb{C}^n$ and the sequence of these random walks for $n \in \mathbb{N}$. More precisely, let us denote by $S_2^n$ the set of triangles with vertices in $V_n$ and edge lengths $2\epsilon_n$, and $C_1^n$ the set of chains associated with this triangulation.

The generator of our random walk in (15), that we reformulate here for the cycle random walk on the torus, is given, for $\sigma \in C_1^n$ and for $F$ a continuous and bounded test function from $C_1^n$ to $\mathbb{R}$, by:

$$A_n F(\sigma) = \epsilon_n^{-2} \sum_{\tau \in S_2^n} (F(\sigma - \partial_2 \tau) - F(\sigma)) w(\sigma, \partial_2 \tau). \quad (37)$$

A convenient class of test functions will be precised later in (44). Let us explain this generator. The rescaling of space is automatic since the number of points on the torus increases and the distance between two connected point is $2\epsilon_n$ which tends to zero. Starting from a cycle $\sigma$, we look at every (oriented) triangle that is adjacent to the cycle. The weight $w(\sigma, \partial_2\tau)$ defined in (23) indicates how many edges (with multiplicity) are shared by $\eta$ and $\tau$. At rate $w(\sigma, \partial_2\tau)$, the triangle $\tau$ (of area in $\epsilon_n^2$) is chosen and its boundary is subtracted to the cycle, entailing the deletion of the shared edges and the inclusion of the new other edges so that the path remains a cycle.

Time is accelerated in $\epsilon_n^{-2}$, which can be understood from the estimate of Proposition 25 below. This explains the $\epsilon_n^{-2}$ at the beginning of the expression of $A_n$.

In this example, $H_1$ is of dimension 2, and with the notation of Theorem 22, we can choose as a basis of $H_1$:

$$\sigma_1 = \sum_{k=0}^{n-2} [(2k\epsilon_n, 0), (2(k+1)\epsilon_n, 0)], \quad \sigma_2 = \sum_{l=0}^{n-2} [(l\epsilon_n, l\sqrt{3}\epsilon_n), ((l+1)\epsilon_n, (l+1)\sqrt{3}\epsilon_n)].$$

Notice that if the initial condition $X_0^n$ is of the form (35) with $\mu_1 > 0$ or $\mu_2 > 0$, then the condition (34) is not satisfied and the state space might be infinite (even if the set of vertices is finite), since a cycle can “loop” on itself and the edge weights can be more than 1 as seen on Fig. 4.
and this is why we focus here on a particular case: we consider the scaling limits of the cycle random walk on the triangulation of the flat torus.

We denote by $\mathbb{T}_2$ the flat torus, which we embed into $\mathbb{R}^2$ as the rectangle $[0, 2] \times [0, \sqrt{3}]$ where the opposite edges are identified. Let $\epsilon_n = 1/n$ and consider

$$V_n = \{(2k\epsilon_n, 2l\sqrt{3}\epsilon_n), 0 \le k < n, 0 \le 2l < n\} \cup \{((2k+1)\epsilon_n, (2l+1)\sqrt{3}\epsilon_n), 0 < 2k+1 \le 2n, 0 < 2l+1 \le n\},$$

the set of vertices of the regular triangulation of mesh $2\epsilon_n$ and denote by $\mathcal{T}_n$ the triangulation based on $V_n$, see Figure 6. We consider the simplicial complex $\mathbb{C}^n$ composed of all the triangles of $\mathcal{T}_n$, their edges and their vertices $V_n$.

<figure><img src="image_9.png" /><figcaption>Figure 6: Regular triangulation of the flat torus. $0 := (0,0)$, $1 := (2\epsilon_n, 0)$, $2 := (\epsilon_n, \sqrt{3}\epsilon_n)$, $3 := (\epsilon_n, -\sqrt{3}\epsilon_n)$, $4 := (-\epsilon_n, \sqrt{3}\epsilon_n)$</figcaption></figure>

We will study here the cycle random walk $(X_t^n)_{t \ge 0}$ (correctly renormalized, as will be seen in the sequel) on this simplicial complex $\mathbb{C}^n$ and the sequence of these random walks for $n \in \mathbb{N}$. More precisely, let us denote by $S_2^n$ the set of triangles with vertices in $V_n$ and edge lengths $2\epsilon_n$, and $C_1^n$ the set of chains associated with this triangulation.

The generator of our random walk in (15), that we reformulate here for the cycle random walk on the torus, is given, for $\sigma \in C_1^n$ and for $F$ a continuous and bounded test function from $C_1^n$ to $\mathbb{R}$, by:

$$A_n F(\sigma) = \epsilon_n^{-2} \sum_{\tau \in S_2^n} (F(\sigma - \partial_2 \tau) - F(\sigma)) w(\sigma, \partial_2 \tau). \quad (37)$$

A convenient class of test functions will be precised later in (44). Let us explain this generator. The rescaling of space is automatic since the number of points on the torus increases and the distance between two connected point is $2\epsilon_n$ which tends to zero. Starting from a cycle $\sigma$, we look at every (oriented) triangle that is adjacent to the cycle. The weight $w(\sigma, \partial_2\tau)$ defined in (23) indicates how many edges (with multiplicity) are shared by $\eta$ and $\tau$. At rate $w(\sigma, \partial_2\tau)$, the triangle $\tau$ (of area in $\epsilon_n^2$) is chosen and its boundary is subtracted to the cycle, entailing the deletion of the shared edges and the inclusion of the new other edges so that the path remains a cycle.

Time is accelerated in $\epsilon_n^{-2}$, which can be understood from the estimate of Proposition 25 below. This explains the $\epsilon_n^{-2}$ at the beginning of the expression of $A_n$.

In this example, $H_1$ is of dimension 2, and with the notation of Theorem 22, we can choose as a basis of $H_1$:

$$\sigma_1 = \sum_{k=0}^{n-2} [(2k\epsilon_n, 0), (2(k+1)\epsilon_n, 0)], \quad \sigma_2 = \sum_{l=0}^{n-2} [(l\epsilon_n, l\sqrt{3}\epsilon_n), ((l+1)\epsilon_n, (l+1)\sqrt{3}\epsilon_n)].$$

Notice that if the initial condition $X_0^n$ is of the form (35) with $\mu_1 > 0$ or $\mu_2 > 0$, then the condition (34) is not satisfied and the state space might be infinite (even if the set of vertices is finite), since a cycle can "loop" on itself and the edge weights can be more than 1 as seen on Fig. 4.
## 4.1 Spectral properties of $L_1^\uparrow$

Recall that $\mathbb{C}^n$ is simplicial complex that is explored and $S_1(\mathbb{C}^n)$ is its set of edges. Before studying the cycle random walk on the triangulation of the 2-dimensional torus, let us give some insight into the spectral properties of $L_1^\uparrow$. To do this, we use the simple edge-valued random walk jumping between upper adjacent edges similar to the one defined in Parzanchevski and Rosenthal [43, 44]. The transition kernel of this random walk is

$$ \forall e, e' \in S_1(\mathbb{C}^n), K(e, e') = \begin{cases} \frac{1}{5} & \text{if } e = -e' \\ \frac{1}{5} & \text{if } e \neq e' \text{ and } e \text{ is upper adjacent to } -e' \\ 0 & \text{otherwise.} \end{cases} $$

Or, equivalently,

$$ \forall e, e' \in S_1(\mathbb{C}^n), K(e, e') = \frac{1}{5} \langle (I - L_1^\uparrow)e^*, e' \rangle_{C^k, C_k} $$

Using standard analysis tool on this random walk gives the following result.

**Proposition 25.** Let $\lambda_m^n$ be the smallest non-zero eigenvalue of the 1-dimensional up-Laplacian on the simplicial complex $\mathbb{C}^n$. We have

$$ \lambda_m^n = \mathcal{O}(\epsilon_n^2). $$

*Proof.* By Corollary 1 in Diaconis and Stroock [14], we have that the smallest non-zero eigenvalue of $L_1^\uparrow$, denoted $\lambda_m^n$, is bounded from below by

$$ \lambda_m^n \geq \frac{2|S_1^+(\mathbb{C}^n)|}{25d^n b^n}, $$

where $d^n$ is the diameter of the state space and $b^n$ is the maximum number of geodesic paths going through a given edge. Since we are triangulating the 2-dimensional torus, we have that

$$ |S_1^+(\mathbb{C}^n)| = \mathcal{O}(\epsilon_n^{-2}) \quad \text{and} \quad d^n = \mathcal{O}(\epsilon_n^{-1}). $$

In order to bound $b^n$, let us note that the total number of geodesic path is $|S_1^+(\mathbb{C}^n)|^2$ while their maximum length is $d^n$. Thus, the sum of the lengths of all geodesic paths is of order $d^n|S_1^+(\mathbb{C}^n)|^2$. Finally, since our triangulation is symmetric the geodesic paths must be distributed among all edges, therefore

$$ b^n = \mathcal{O}\left(\frac{d^n|S_1^+(\mathbb{C}^n)|^2}{|S_1^+(\mathbb{C}^n)|}\right) = \mathcal{O}(\epsilon_n^{-3}), $$

which concludes the proof.

The result of Proposition 25 gives us indications on the rescaling needed for our random walk to converge to some continuous limit. Next, we investigate the state space in which the convergence should be established.

## 4.2 From chains to currents

In the same way that $V_n$ was embedded into $\mathbb{T}_2$, we need to embed the spaces of 1-chains into a bigger continuous space. The natural proposition is to consider chains as *currents*, which are duals of linear forms on the torus. We first recall some notation and refer to [5, Chapters 5 and 6], especially on differential forms and their integrals. Recall also that we focus in this section on 1-chains and thus will not need to introduce $k$-forms in their whole generality, but only forms for $k \in \{0, 1, 2\}$. Denote by $T_x^*\mathbb{T}_2$ the tangent vector space on the torus at $x$. The spaces $\Lambda^k T_x^*\mathbb{T}_2$ are the spaces of alternating $k$-linear forms on $T_x^*\mathbb{T}_2$. Note that $\Lambda^0 T_x^*\mathbb{T}_2 = \mathbb{R}$ (so that 0-forms are continuous real functions on the torus), $\Lambda^1 T_x^*\mathbb{T}_2 = T_x^*\mathbb{T}_2$. For $k = 2$, $\Lambda^2 T_x^*\mathbb{T}_2$ is the set of bilinear real functions $\phi$ on $T_x^*\mathbb{T}_2$, such that $\forall u_1, u_2 \in T_x^*\mathbb{T}_2, \phi(u_1, u_2) = -\phi(u_2, u_1)$. It is then classical to define by $T^*\mathbb{T}_2 = \cup_{x \in \mathbb{T}_2} T_x^*\mathbb{T}_2$ the tangent bundle of the torus, and to let $\Lambda^k T^*\mathbb{T}_2 = \cup_{x \in \mathbb{T}_2} \Lambda^k T_x^*\mathbb{T}_2$ be the disjoint union of the spaces $\Lambda^k T_x^*\mathbb{T}_2$.

We denote by $\mathcal{C}^{k,p}(\mathbb{T}_2) = \mathcal{C}^p(\mathbb{T}_2, \Lambda^k T^*\mathbb{T}_2)$ the set of differential $k$-form of class $\mathcal{C}^p$ on $\mathbb{T}_2$. The space of continuous 1-differential forms is denoted by $\mathcal{C}^1(\mathbb{T}_2)$ and $\mathcal{C}^{1,p}(\mathbb{T}_2)$ is dense in $\mathcal{C}^1(\mathbb{T}_2)$. Notice that $\mathcal{C}^1(\mathbb{T}_2)$ is separable (see [5, Th.5.1.5 P.147]).
## 4.1 Spectral properties of $L_1^\uparrow$

Recall that $\mathbb{C}^n$ is simplicial complex that is explored and $S_1(\mathbb{C}^n)$ is its set of edges. Before studying the cycle random walk on the triangulation of the 2-dimensional torus, let us give some insight into the spectral properties of $L_1^\uparrow$. To do this, we use the simple edge-valued random walk jumping between upper adjacent edges similar to the one defined in Parzanchevski and Rosenthal [43, 44]. The transition kernel of this random walk is

$$ \forall e, e' \in S_1(\mathbb{C}^n), K(e, e') = \begin{cases} \frac{1}{5} & \text{if } e = -e' \\ \frac{1}{5} & \text{if } e \neq e' \text{ and } e \text{ is upper adjacent to } -e' \\ 0 & \text{otherwise.} \end{cases} $$

Or, equivalently,

$$ \forall e, e' \in S_1(\mathbb{C}^n), K(e, e') = \frac{1}{5} \langle (I - L_1^\uparrow)e^*, e' \rangle_{C^k, C_k} $$

Using standard analysis tool on this random walk gives the following result.

**Proposition 25.** Let $\lambda_m^n$ be the smallest non-zero eigenvalue of the 1-dimensional up-Laplacian on the simplicial complex $\mathbb{C}^n$. We have

$$ \lambda_m^n = \mathcal{O}(\epsilon_n^2). $$

*Proof.* By Corollary 1 in Diaconis and Stroock [14], we have that the smallest non-zero eigenvalue of $L_1^\uparrow$, denoted $\lambda_m^n$, is bounded from below by

$$ \lambda_m^n \geq \frac{2|S_1^+(\mathbb{C}^n)|}{25d^n b^n}, $$

where $d^n$ is the diameter of the state space and $b^n$ is the maximum number of geodesic paths going through a given edge. Since we are triangulating the 2-dimensional torus, we have that

$$ |S_1^+(\mathbb{C}^n)| = \mathcal{O}(\epsilon_n^{-2}) \quad \text{and} \quad d^n = \mathcal{O}(\epsilon_n^{-1}). $$

In order to bound $b^n$, let us note that the total number of geodesic path is $|S_1^+(\mathbb{C}^n)|^2$ while their maximum length is $d^n$. Thus, the sum of the lengths of all geodesic paths is of order $d^n|S_1^+(\mathbb{C}^n)|^2$. Finally, since our triangulation is symmetric the geodesic paths must be distributed among all edges, therefore

$$ b^n = \mathcal{O} \left( \frac{d^n |S_1^+(\mathbb{C}^n)|^2}{|S_1^+(\mathbb{C}^n)|} \right) = \mathcal{O}(\epsilon_n^{-3}), $$

which concludes the proof.

The result of Proposition 25 gives us indications on the rescaling needed for our random walk to converge to some continuous limit. Next, we investigate the state space in which the convergence should be established.

## 4.2 From chains to currents

In the same way that $V_n$ was embedded into $\mathbb{T}_2$, we need to embed the spaces of 1-chains into a bigger continuous space. The natural proposition is to consider chains as *currents*, which are duals of linear forms on the torus. We first recall some notation and refer to [5, Chapters 5 and 6], especially on differential forms and their integrals. Recall also that we focus in this section on 1-chains and thus will not need to introduce $k$-forms in their whole generality, but only forms for $k \in \{0, 1, 2\}$. Denote by $T_x^*\mathbb{T}_2$ the tangent vector space on the torus at $x$. The spaces $\Lambda^k T_x^*\mathbb{T}_2$ are the spaces of alternating $k$-linear forms on $T_x^*\mathbb{T}_2$. Note that $\Lambda^0 T_x^*\mathbb{T}_2 = \mathbb{R}$ (so that 0-forms are continuous real functions on the torus), $\Lambda^1 T_x^*\mathbb{T}_2 = T_x^*\mathbb{T}_2$. For $k = 2$, $\Lambda^2 T_x^*\mathbb{T}_2$ is the set of bilinear real functions $\phi$ on $T_x^*\mathbb{T}_2$, such that $\forall u_1, u_2 \in T_x^*\mathbb{T}_2, \phi(u_1, u_2) = -\phi(u_2, u_1)$. It is then classical to define by $T^*\mathbb{T}_2 = \cup_{x \in \mathbb{T}_2} T_x^*\mathbb{T}_2$ the tangent bundle of the torus, and to let $\Lambda^k T^*\mathbb{T}_2 = \cup_{x \in \mathbb{T}_2} \Lambda^k T_x^*\mathbb{T}_2$ be the disjoint union of the spaces $\Lambda^k T_x^*\mathbb{T}_2$.

We denote by $\mathcal{C}^{k,p}(\mathbb{T}_2) = \mathcal{C}^p(\mathbb{T}_2, \Lambda^k T^*\mathbb{T}_2)$ the set of differential $k$-form of class $\mathcal{C}^p$ on $\mathbb{T}_2$. The space of continuous 1-differential forms is denoted by $\mathcal{C}^1(\mathbb{T}_2)$ and $\mathcal{C}^{1,p}(\mathbb{T}_2)$ is dense in $\mathcal{C}^1(\mathbb{T}_2)$. Notice that $\mathcal{C}^1(\mathbb{T}_2)$ is separable (see [5, Th.5.1.5 P.147]).
**Definition 26.** Let us denote by $dx_1$ and $dx_2$ the canonical basis of the space of linear forms on $T^*T_2$. For a continuous 1-differential form $\phi \in \mathfrak{C}^1(T_2)$, we can write

$$ \phi = \phi^1 \, dx_1 + \phi^2 \, dx_2, \qquad (38) $$

where $\phi^1$ and $\phi^2$ are continuous functions on the torus, i.e. they can be viewed as the restriction over $[0, 1]^2$ of continuous (1, 1)-periodic functions:

$$ \phi(x_1 + l_1, x_2 + l_2) = \phi(x_1, x_2) $$

for any pair of integers $(l_1, l_2) \in \mathbb{Z}^2$. We set

$$ \|\phi\|_{\mathfrak{C}^1} = \|\phi^1\|_{\infty} + \|\phi^2\|_{\infty}. $$

It topological dual is the set of currents, denoted by $\mathfrak{C}_1$. It inherits the Banach norm:

$$ \|p\|_{\mathfrak{C}_1} = \sup_{\phi \in \mathfrak{C}^1} \frac{|\langle p, \phi \rangle_{\mathfrak{C}_1, \mathfrak{C}^1}|}{\|\phi\|_{\mathfrak{C}^1}}. \qquad (39) $$

By the previous definition and the preceding remarks, $(\mathfrak{C}_1, \|\cdot\|_{\mathfrak{C}_1})$ is a Polish space. Recall that for two functions $\phi$ and $\varphi \in T_x^*T_2$, $\phi \wedge \varphi \in \Lambda^2 T_x^*T_2$ is defined for all $u_1, u_2 \in T_x^*T_2$ by:

$$ \phi \wedge \varphi(u_1, u_2) = \phi(u_1)\varphi(u_2) - \varphi(u_1)\phi(u_2). \qquad (40) $$

In particular, the 2-form $dx_1 \wedge dx_2$ is a volume form on $T_2$ that is canonically associated with the Lebesgue measure on $T_2$ and allows to integrate differential forms (see Chapter 6 of [5], for example).

As a particular case of application, let us mention the following useful definition:

**Definition 27.** We denote by $\mathfrak{P}$, the set of paths, i.e. the piecewise differentiable maps from $[0, 1]$ into $T_2$. This set can be viewed as a subspace of $\mathfrak{C}^1$. For $\phi \in \mathfrak{C}_1$, the curvilinear integral of $\phi$ along an element $p \in \mathfrak{P}$ is a linear map and we denote by $\langle p, \phi \rangle$ the integral $\int_p \phi$, with

$$ |\langle p, \phi \rangle| = \left| \int_p \phi \right| \le \|\phi\|_{\mathfrak{C}_1} \|p\|_{\mathfrak{C}^1}. $$

The definition 27 allows us to see 1-chains as paths by embedding the abstract graph into a geometric graph on the torus.

In particular, for a cycle $\sigma$ constructed on $\mathcal{T}_n$,

$$ |\langle \sigma, \phi \rangle_{\mathfrak{C}_1, \mathfrak{C}^1}| = \left| \int_{\sigma} \phi \right| \le \|\phi\|_{\mathfrak{C}^1} 2\epsilon_n \sum_{e \in \mathcal{T}_n} |\lambda_e(\sigma)|, \qquad (41) $$

with the notation of (3). Since the $\lambda_e(\sigma)$ are integers, the right hand side is upper-bounded by $\|\sigma\|_{\mathfrak{C}_1}^2$ (in discrete norm). As a consequence:

$$ \|\sigma\|_{\mathfrak{C}_1} \le 2\epsilon_n \|\sigma\|_{\mathfrak{C}_1}^2. \qquad (42) $$

Notice that when $\lambda_e \in \{-1, 0, 1\}$, the right hand side of (41) is equal to $2\epsilon_n \|\sigma\|_{\mathfrak{C}_1}^2$, which is the metric length of the chain $\sigma$.

Keeping in mind Donsker's theorem, we expect that the length of the diffusive limit cycle is infinite in the same way as the standard Brownian motion has infinite variation. Therefore, we will also require for what follows additional regularity assumptions on the differential forms.

Recall that $\mathfrak{C}^{1,p}$ denotes the set of 1-differential forms of class $C^p$, i.e. for which the functions $\phi^1$ and $\phi^2$ in (38) are $C^p$. In the sequel, we will consider $p \in \{1, 2, 3\}$. Since we are on the compact torus, these derivatives are bounded.
**Definition 26.** Let us denote by $dx_1$ and $dx_2$ the canonical basis of the space of linear forms on $T^*T_2$. For a continuous 1-differential form $\phi \in \mathcal{C}^1(T_2)$, we can write

$$ \phi = \phi^1 \, dx_1 + \phi^2 \, dx_2, \qquad (38) $$

where $\phi^1$ and $\phi^2$ are continuous functions on the torus, i.e. they can be viewed as the restriction over $[0, 1]^2$ of continuous (1, 1)-periodic functions:

$$ \phi(x_1 + l_1, x_2 + l_2) = \phi(x_1, x_2) $$

for any pair of integers $(l_1, l_2) \in \mathbb{Z}^2$. We set

$$ \|\phi\|_{\mathcal{C}^1} = \|\phi^1\|_{\infty} + \|\phi^2\|_{\infty}. $$

It topological dual is the set of currents, denoted by $\mathcal{C}_1$. It inherits the Banach norm:

$$ \|p\|_{\mathcal{C}_1} = \sup_{\phi \in \mathcal{C}^1} \frac{|\langle p, \phi \rangle_{\mathcal{C}_1, \mathcal{C}^1}|}{\|\phi\|_{\mathcal{C}^1}}. \qquad (39) $$

By the previous definition and the preceding remarks, $(\mathcal{C}_1, \|\cdot\|_{\mathcal{C}_1})$ is a Polish space. Recall that for two functions $\phi$ and $\varphi \in T_x^*T_2$, $\phi \wedge \varphi \in \Lambda^2 T_x^*T_2$ is defined for all $u_1, u_2 \in T_x^*T_2$ by:

$$ \phi \wedge \varphi(u_1, u_2) = \phi(u_1)\varphi(u_2) - \varphi(u_1)\phi(u_2). \qquad (40) $$

In particular, the 2-form $dx_1 \wedge dx_2$ is a volume form on $T_2$ that is canonically associated with the Lebesgue measure on $T_2$ and allows to integrate differential forms (see Chapter 6 of [5], for example).

As a particular case of application, let us mention the following useful definition:

**Definition 27.** We denote by $\mathfrak{P}$, the set of paths, i.e. the piecewise differentiable maps from $[0, 1]$ into $T_2$. This set can be viewed as a subspace of $\mathcal{C}^1$. For $\phi \in \mathcal{C}_1$, the curvilinear integral of $\phi$ along an element $p \in \mathfrak{P}$ is a linear map and we denote by $\langle p, \phi \rangle$ the integral $\int_p \phi$, with

$$ |\langle p, \phi \rangle| = \left| \int_p \phi \right| \le \|\phi\|_{\mathcal{C}_1} \|p\|_{\mathcal{C}^1}. $$

The definition 27 allows us to see 1-chains as paths by embedding the abstract graph into a geometric graph on the torus.

In particular, for a cycle $\sigma$ constructed on $\mathcal{T}_n$,

$$ |\langle \sigma, \phi \rangle_{\mathcal{C}_1, \mathcal{C}^1}| = \left| \int_{\sigma} \phi \right| \le \|\phi\|_{\mathcal{C}^1} 2\epsilon_n \sum_{e \in \mathcal{T}_n} |\lambda_e(\sigma)|, \qquad (41) $$

with the notation of (3). Since the $\lambda_e(\sigma)$ are integers, the right hand side is upper-bounded by $\|\sigma\|_{\mathcal{C}_1}^2$ (in discrete norm). As a consequence:

$$ \|\sigma\|_{\mathcal{C}_1} \le 2\epsilon_n \|\sigma\|_{\mathcal{C}_1}^2. \qquad (42) $$

Notice that when $\lambda_e \in \{-1, 0, 1\}$, the right hand side of (41) is equal to $2\epsilon_n \|\sigma\|_{\mathcal{C}_1}^2$, which is the metric length of the chain $\sigma$.

Keeping in mind Donsker's theorem, we expect that the length of the diffusive limit cycle is infinite in the same way as the standard Brownian motion has infinite variation. Therefore, we will also require for what follows additional regularity assumptions on the differential forms.

Recall that $\mathcal{C}^{1,p}$ denotes the set of 1-differential forms of class $C^p$, i.e. for which the functions $\phi^1$ and $\phi^2$ in (38) are $C^p$. In the sequel, we will consider $p \in \{1, 2, 3\}$. Since we are on the compact torus, these derivatives are bounded.
**Definition 28.** Let us embed $\mathcal{C}^{1,3}$ with the norm:

$$ \|\phi\|_{\mathcal{C}^{1,3}} = \|\phi\|_{\mathcal{C}^1} + \sum_{i=1}^{2} \left( \left\| \frac{\partial \phi^i}{\partial x_1} \right\|_{\infty} + \left\| \frac{\partial \phi^i}{\partial x_2} \right\|_{\infty} \right. \\ \left. + \left\| \frac{\partial^2 \phi^i}{\partial x_1^2} \right\|_{\infty} + \left\| \frac{\partial^2 \phi^i}{\partial x_2^2} \right\|_{\infty} + \left\| \frac{\partial^2 \phi^i}{\partial x_1 \partial x_2} \right\|_{\infty} + \left\| \frac{\partial^3 \phi^i}{\partial x_1^3} \right\|_{\infty} + \left\| \frac{\partial^3 \phi^i}{\partial x_1^2 \partial x_2} \right\|_{\infty} + \left\| \frac{\partial^3 \phi^i}{\partial x_1 \partial x_2^2} \right\|_{\infty} + \left\| \frac{\partial^3 \phi^i}{\partial x_2^3} \right\|_{\infty} \right). $$

Its topological dual is denoted by $\mathcal{C}_{1,3}$ and embedded with the Banach norm:

$$ \|\sigma\|_{\mathcal{C}_{1,3}} = \sup_{\phi \in \mathcal{C}_{1,3}} \frac{|\langle \sigma, \phi \rangle_{\mathcal{C}_{1,3}, \mathcal{C}^{1,3}}|}{\|\phi\|_{\mathcal{C}^{1,3}}}. \quad (43) $$

The space $(\mathcal{C}_{1,3}, \|\cdot\|_{\mathcal{C}_{1,3}})$ is Polish and contains $\mathcal{C}_1$ and $\mathfrak{P}$. Remark that if $\sigma \in \mathcal{C}_1$, then the bracket in the numerator of the right hand side of (43) is equal to $\langle \sigma, \phi \rangle_{\mathcal{C}_1, \mathcal{C}^1}$. In the sequel, we are interested in limit theorems for the sequence of cycle random walks viewed as a $\mathcal{C}_{1,3}$-valued random process.

A function $F: \mathcal{C}_{1,3} \to \mathbb{R}$ is said to be cylindrical if it is of the form:

$$ F(\sigma) = f(\langle \sigma, \phi_1 \rangle_{\mathcal{C}_{1,3}, \mathcal{C}^{1,3}}, \dots, \langle \sigma, \phi_k \rangle_{\mathcal{C}_{1,3}, \mathcal{C}^{1,3}}) \quad (44) $$

for some $\phi_1, \dots, \phi_k \in \mathcal{C}^{1,3}$ and $f$ a continuous real function of class $\mathcal{C}^1$ with compact support from $\mathbb{R}^k$ into $\mathbb{R}$. Let $\mathcal{B}$ be the Banach space of bounded continuous functions from $\mathcal{C}_{1,3}$ to $\mathbb{R}$ equipped with the sup-norm. The set of cylindrical functions is a set of continuous functions on $\mathcal{C}_{1,3}$ that is closed under addition and separates points $\mathcal{C}_{1,3}$. For the latter point, we can notice with arguments similar to [36, Lemma A.1] that the topology generated on $\mathcal{C}_{1,3}$ by these functions is the same as the one of finite-dimensional convergence. These properties will be useful in the next section, when establishing the convergence of the cycle-random walk on the triangulation of the torus supported on $V_n$.

## 4.3 Hodge operator

For $0 \le k \le 1$ (since we work on the 2d-torus) and $p \ge 1$, there exists a unique linear operator $d$ from $\mathcal{C}^{k,p}(\mathbb{T}_2)$ into $\mathcal{C}^{k+1,p-1}(\mathbb{T}_2)$, called exterior differentiation (see [5, Prop. 5.2.9.1]), such that:

(i) $d \circ d = 0$

(ii) for $k$ and $k'$-forms $\phi$ and $\varphi$, $d(\phi \wedge \varphi) = d\phi \wedge \varphi + (-1)^k \phi \wedge d\varphi$,

(iii) for $\phi \in \mathcal{C}^1(\mathbb{T}_2, \mathbb{R})$, $d\phi : X \to T^*X$ is the derivative of $\phi$:

$$ d\phi(x) = \frac{\partial \phi}{\partial x_1}(x)dx_1 + \frac{\partial \phi}{\partial x_2}(x)dx_2. \quad (45) $$

For $\phi \in \mathcal{C}^{1,1}(\mathbb{T}_2)$, there exists two $\mathcal{C}^1(\mathbb{T}_2, \mathbb{R})$ functions $\phi_1$ and $\phi_2$ such that $\phi(x) = \phi^1(x) dx_1 + \phi^2(x) dx_2$. The exterior derivative of $\phi$, $d\phi \in \mathcal{C}^{2,0}(\mathbb{T}_2)$, is then:

$$ \begin{aligned} d\phi &= d\phi^1 \wedge dx_1 + d\phi^2 \wedge dx_2 \\ &= \frac{\partial \phi^1}{\partial x_2} dx_2 \wedge dx_1 + \frac{\partial \phi^2}{\partial x_1} dx_1 \wedge dx_2 \\ &= \left( \frac{\partial \phi^2}{\partial x_1}(x) - \frac{\partial \phi^1}{\partial x_2}(x) \right) dx_1 \wedge dx_2. \end{aligned} \quad (46) $$

**Definition 29.** The Hodge transform of forms is the linear transformation defined by its action on a basis of differential forms:

$$ *1 = dx_1 \wedge dx_2, \quad * dx_1 = dx_2, \quad * dx_2 = - dx_1, \quad *( dx_1 \wedge dx_2) = 1. $$

**Definition 30.** The Rham-Hodge operator is then defined by

$$ \mathcal{L} = \mathcal{L}^{\uparrow} + \mathcal{L}^{\downarrow} \text{ where } \mathcal{L}^{\uparrow} = *d * d \text{ and } \mathcal{L}^{\downarrow} = d * d * . $$
**Definition 28.** Let us embed $\mathcal{C}^{1,3}$ with the norm:

$$ \|\phi\|_{\mathcal{C}^{1,3}} = \|\phi\|_{\mathcal{C}^1} + \sum_{i=1}^{2} \left( \left\| \frac{\partial \phi^i}{\partial x_1} \right\|_{\infty} + \left\| \frac{\partial \phi^i}{\partial x_2} \right\|_{\infty} \right. \\ \left. + \left\| \frac{\partial^2 \phi^i}{\partial x_1^2} \right\|_{\infty} + \left\| \frac{\partial^2 \phi^i}{\partial x_2^2} \right\|_{\infty} + \left\| \frac{\partial^2 \phi^i}{\partial x_1 \partial x_2} \right\|_{\infty} + \left\| \frac{\partial^3 \phi^i}{\partial x_1^3} \right\|_{\infty} + \left\| \frac{\partial^3 \phi^i}{\partial x_1^2 \partial x_2} \right\|_{\infty} + \left\| \frac{\partial^3 \phi^i}{\partial x_1 \partial x_2^2} \right\|_{\infty} + \left\| \frac{\partial^3 \phi^i}{\partial x_2^3} \right\|_{\infty} \right). $$

Its topological dual is denoted by $\mathcal{C}_{1,3}$ and embedded with the Banach norm:

$$ \|\sigma\|_{\mathcal{C}_{1,3}} = \sup_{\phi \in \mathcal{C}_{1,3}} \frac{|\langle \sigma, \phi \rangle_{\mathcal{C}_{1,3}, \mathcal{C}^{1,3}}|}{\|\phi\|_{\mathcal{C}^{1,3}}}. \quad (43) $$

The space $(\mathcal{C}_{1,3}, \|\cdot\|_{\mathcal{C}_{1,3}})$ is Polish and contains $\mathcal{C}_1$ and $\mathfrak{P}$. Remark that if $\sigma \in \mathcal{C}_1$, then the bracket in the numerator of the right hand side of (43) is equal to $\langle \sigma, \phi \rangle_{\mathcal{C}_1, \mathcal{C}^1}$. In the sequel, we are interested in limit theorems for the sequence of cycle random walks viewed as a $\mathcal{C}_{1,3}$-valued random process.

A function $F: \mathcal{C}_{1,3} \to \mathbb{R}$ is said to be cylindrical if it is of the form:

$$ F(\sigma) = f(\langle \sigma, \phi_1 \rangle_{\mathcal{C}_{1,3}, \mathcal{C}^{1,3}}, \dots, \langle \sigma, \phi_k \rangle_{\mathcal{C}_{1,3}, \mathcal{C}^{1,3}}) \quad (44) $$

for some $\phi_1, \dots, \phi_k \in \mathcal{C}^{1,3}$ and $f$ a continuous real function of class $\mathcal{C}^1$ with compact support from $\mathbb{R}^k$ into $\mathbb{R}$. Let $\mathcal{B}$ be the Banach space of bounded continuous functions from $\mathcal{C}_{1,3}$ to $\mathbb{R}$ equipped with the sup-norm. The set of cylindrical functions is a set of continuous functions on $\mathcal{C}_{1,3}$ that is closed under addition and separates points $\mathcal{C}_{1,3}$. For the latter point, we can notice with arguments similar to [36, Lemma A.1] that the topology generated on $\mathcal{C}_{1,3}$ by these functions is the same as the one of finite-dimensional convergence. These properties will be useful in the next section, when establishing the convergence of the cycle-random walk on the triangulation of the torus supported on $V_n$.

## 4.3 Hodge operator

For $0 \le k \le 1$ (since we work on the 2d-torus) and $p \ge 1$, there exists a unique linear operator $d$ from $\mathcal{C}^{k,p}(\mathbb{T}_2)$ into $\mathcal{C}^{k+1,p-1}(\mathbb{T}_2)$, called exterior differentiation (see [5, Prop. 5.2.9.1]), such that:

(i) $d \circ d = 0$

(ii) for $k$ and $k'$-forms $\phi$ and $\varphi$, $d(\phi \wedge \varphi) = d\phi \wedge \varphi + (-1)^k \phi \wedge d\varphi$,

(iii) for $\phi \in \mathcal{C}^1(\mathbb{T}_2, \mathbb{R})$, $d\phi : X \to T^*X$ is the derivative of $\phi$:

$$ d\phi(x) = \frac{\partial \phi}{\partial x_1}(x)dx_1 + \frac{\partial \phi}{\partial x_2}(x)dx_2. \quad (45) $$

For $\phi \in \mathcal{C}^{1,1}(\mathbb{T}_2)$, there exists two $\mathcal{C}^1(\mathbb{T}_2, \mathbb{R})$ functions $\phi_1$ and $\phi_2$ such that $\phi(x) = \phi^1(x) dx_1 + \phi^2(x) dx_2$. The exterior derivative of $\phi$, $d\phi \in \mathcal{C}^{2,0}(\mathbb{T}_2)$, is then:

$$ \begin{aligned} d\phi &= d\phi^1 \wedge dx_1 + d\phi^2 \wedge dx_2 \\ &= \frac{\partial \phi^1}{\partial x_2} dx_2 \wedge dx_1 + \frac{\partial \phi^2}{\partial x_1} dx_1 \wedge dx_2 \\ &= \left( \frac{\partial \phi^2}{\partial x_1}(x) - \frac{\partial \phi^1}{\partial x_2}(x) \right) dx_1 \wedge dx_2. \end{aligned} \quad (46) $$

**Definition 29.** The Hodge transform of forms is the linear transformation defined by its action on a basis of differential forms:

$$ *1 = dx_1 \wedge dx_2, \quad * dx_1 = dx_2, \quad * dx_2 = - dx_1, \quad *( dx_1 \wedge dx_2) = 1. $$

**Definition 30.** The Rham-Hodge operator is then defined by

$$ \mathcal{L} = \mathcal{L}^{\uparrow} + \mathcal{L}^{\downarrow} \text{ where } \mathcal{L}^{\uparrow} = *d * d \text{ and } \mathcal{L}^{\downarrow} = d * d * . $$
**Proposition 31.** Let $\phi \in \mathcal{C}^{1,2}(\mathbb{T}_2)$, with $\phi = \phi^1 dx_1 + \phi^2 dx_2$. We have

$$ \mathcal{L}^{\uparrow}(\phi) = (\phi_{22}^1 - \phi_{12}^1) dx_1 + (\phi_{11}^2 - \phi_{12}^1) dx_2 \quad (47) $$

$$ \mathcal{L}^{\downarrow}(\phi) = (\phi_{11}^1 + \phi_{12}^2) dx_1 + (\phi_{12}^1 + \phi_{22}^2) dx_2. \quad (48) $$

where $f_i$ is a shortcut for the partial derivative of $f$ with respect to the variable $x_i$.

*Proof.* Using Def. 29 and (46),

$$ *d\phi = \frac{\partial\phi^2}{\partial x_1} - \frac{\partial\phi^1}{\partial x_2}. \quad (49) $$

Then, by (45),

$$ \begin{aligned} d * d\phi &= \left(\frac{\partial^2\phi^2}{\partial x_1^2} - \frac{\partial^2\phi^1}{\partial x_1\partial x_2}\right) dx_1 + \left(\frac{\partial^2\phi^2}{\partial x_1\partial x_2} - \frac{\partial^2\phi^1}{\partial x_2^2}\right) dx_2 \\ &= (\phi_{11}^2 - \phi_{12}^1) dx_1 + (\phi_{12}^2 - \phi_{22}^1) dx_2 \end{aligned} $$

Using again Def. 29 gives (47). Proceeding similarly provides the expression of $\mathcal{L}^{\downarrow}\phi$.

<figure><img src="image_8.png" /></figure>

Notice that the Rham-Hodge operator appears also as the Witten Laplacian in the literature for 0-forms (e.g. [34, 56]), which relates to the generator of diffusions on manifolds [4, 16, 33, 51] – in our case, the Brownian motion on the torus $\mathbb{T}_2$. For 1-forms, [16, 33] provide probabilistic representations (different from ours) of the semigroups associated with the Rham-Hodge operator by mean of Feynman-Kac formulas and using the diffusions obtained for 0-forms.

**Proposition 32.** The operator $\mathcal{L}^{\uparrow}$ is closable, dissipative and there exists $\lambda > 0$ such that $\lambda \mathrm{Id} - \mathcal{L}^{\uparrow}$ is one-to-one. As a consequence, $\mathcal{L}^{\uparrow}$, whose domain contains $\mathcal{C}^{1,2}$, generates a strongly continuous contraction semi-group. Moreover, the martingale problem associated to $\mathcal{L}^{\uparrow}$ is well posed.

*Proof.* First, let us prove that $\mathcal{L}^{\uparrow}$ is closable, dissipative and that there exists $\lambda > 0$ such that $\lambda \mathrm{Id} - \mathcal{L}^{\uparrow}$ is one-to-one.

By (47), for $\phi \in \mathcal{C}^{1,2}$,

$$ \langle \mathcal{L}^{\uparrow}\phi, \phi \rangle = \int_{\mathbb{T}_2} \frac{\partial}{\partial x_2} (\phi_2^1 - \phi_1^2) \phi^1 dx_1 dx_2 - \int_{\mathbb{T}_2} \frac{\partial}{\partial x_1} (\phi_2^1 - \phi_1^2) \phi^2 dx_1 dx_2. $$

By integration by parts, taking into account the periodicity of $\phi^1$ and $\phi^2$, we get

$$ \begin{aligned} \langle \mathcal{L}^{\uparrow}\phi, \phi \rangle &= - \int_{\mathbb{T}_2} (\phi_2^1 - \phi_1^2) \phi_2^1 dx_1 dx_2 + \int_{\mathbb{T}_2} (\phi_2^1 - \phi_1^2) \phi_1^2 dx_1 dx_2 \\ &= - \int_{\mathbb{T}_2} (\phi_2^1 - \phi_1^2)^2 dx_1 dx_2. \end{aligned} $$

This means that $\mathcal{L}^{\uparrow}$ is symmetric and negative (hence dissipative). Integrating by parts a second time yields, for any $\psi \in \mathcal{C}^{1,2}$,

$$ \langle \mathcal{L}^{\uparrow}\phi, \psi \rangle = \langle \phi, \mathcal{L}^{\uparrow}\psi \rangle. $$

Hence, if $\phi_n \to 0$ and $\mathcal{L}^{\uparrow}\phi_n \to \eta$, we get $\langle \eta, \psi \rangle = 0$ for any $\psi \in \mathcal{C}^{1,2}$. By density, this entails $\eta = 0$. Hence, $\mathcal{L}^{\uparrow}$ is closable. We still denote by $\mathcal{L}^{\uparrow}$ its extension, whose domain $\mathrm{dom}(\mathcal{L}^{\uparrow})$ contains at least $\mathcal{C}^{1,2}$. Consider the basis of $L^2(\mathbb{T}_2, \mathbb{C})$ given by

$$ e_{n,m}(x_1, x_2) = e^{2i\pi nx_1} e^{2i\pi mx_2}, \quad n, m \in \mathbb{Z}. $$

For $i = 1, 2$, we have in $L^2(\mathbb{T}_2, \mathbb{C})$,

$$ \phi^i = \sum_{n,m \in \mathbb{Z}} c_{n,m}^i e_{n,m}. $$
**Proposition 31.** Let $\phi \in \mathcal{C}^{1,2}(\mathbb{T}_2)$, with $\phi = \phi^1 dx_1 + \phi^2 dx_2$. We have

$$ \mathcal{L}^{\uparrow}(\phi) = (\phi_{22}^1 - \phi_{12}^1) dx_1 + (\phi_{11}^2 - \phi_{12}^1) dx_2 \quad (47) $$

$$ \mathcal{L}^{\downarrow}(\phi) = (\phi_{11}^1 + \phi_{12}^2) dx_1 + (\phi_{12}^1 + \phi_{22}^2) dx_2. \quad (48) $$

where $f_i$ is a shortcut for the partial derivative of $f$ with respect to the variable $x_i$.

*Proof.* Using Def. 29 and (46),

$$ *d\phi = \frac{\partial\phi^2}{\partial x_1} - \frac{\partial\phi^1}{\partial x_2}. \quad (49) $$

Then, by (45),

$$ \begin{aligned} d * d\phi &= \left(\frac{\partial^2\phi^2}{\partial x_1^2} - \frac{\partial^2\phi^1}{\partial x_1\partial x_2}\right) dx_1 + \left(\frac{\partial^2\phi^2}{\partial x_1\partial x_2} - \frac{\partial^2\phi^1}{\partial x_2^2}\right) dx_2 \\ &= (\phi_{11}^2 - \phi_{12}^1) dx_1 + (\phi_{12}^2 - \phi_{22}^1) dx_2 \end{aligned} $$

Using again Def. 29 gives (47). Proceeding similarly provides the expression of $\mathcal{L}^{\downarrow}\phi$.

<figure><img src="image_10.png" /></figure>

Notice that the Rham-Hodge operator appears also as the Witten Laplacian in the literature for 0-forms (e.g. [34, 56]), which relates to the generator of diffusions on manifolds [4, 16, 33, 51] – in our case, the Brownian motion on the torus $\mathbb{T}_2$. For 1-forms, [16, 33] provide probabilistic representations (different from ours) of the semigroups associated with the Rham-Hodge operator by mean of Feynman-Kac formulas and using the diffusions obtained for 0-forms.

**Proposition 32.** The operator $\mathcal{L}^{\uparrow}$ is closable, dissipative and there exists $\lambda > 0$ such that $\lambda \mathrm{Id} - \mathcal{L}^{\uparrow}$ is one-to-one. As a consequence, $\mathcal{L}^{\uparrow}$, whose domain contains $\mathcal{C}^{1,2}$, generates a strongly continuous contraction semi-group. Moreover, the martingale problem associated to $\mathcal{L}^{\uparrow}$ is well posed.

*Proof.* First, let us prove that $\mathcal{L}^{\uparrow}$ is closable, dissipative and that there exists $\lambda > 0$ such that $\lambda \mathrm{Id} - \mathcal{L}^{\uparrow}$ is one-to-one.

By (47), for $\phi \in \mathcal{C}^{1,2}$,

$$ \langle \mathcal{L}^{\uparrow}\phi, \phi \rangle = \int_{\mathbb{T}_2} \frac{\partial}{\partial x_2} (\phi_2^1 - \phi_1^2) \phi^1 dx_1 dx_2 - \int_{\mathbb{T}_2} \frac{\partial}{\partial x_1} (\phi_2^1 - \phi_1^2) \phi^2 dx_1 dx_2. $$

By integration by parts, taking into account the periodicity of $\phi^1$ and $\phi^2$, we get

$$ \begin{aligned} \langle \mathcal{L}^{\uparrow}\phi, \phi \rangle &= - \int_{\mathbb{T}_2} (\phi_2^1 - \phi_1^2) \phi_2^1 dx_1 dx_2 + \int_{\mathbb{T}_2} (\phi_2^1 - \phi_1^2) \phi_1^2 dx_1 dx_2 \\ &= - \int_{\mathbb{T}_2} (\phi_2^1 - \phi_1^2)^2 dx_1 dx_2. \end{aligned} $$

This means that $\mathcal{L}^{\uparrow}$ is symmetric and negative (hence dissipative). Integrating by parts a second time yields, for any $\psi \in \mathcal{C}^{1,2}$,

$$ \langle \mathcal{L}^{\uparrow}\phi, \psi \rangle = \langle \phi, \mathcal{L}^{\uparrow}\psi \rangle. $$

Hence, if $\phi_n \to 0$ and $\mathcal{L}^{\uparrow}\phi_n \to \eta$, we get $\langle \eta, \psi \rangle = 0$ for any $\psi \in \mathcal{C}^{1,2}$. By density, this entails $\eta = 0$. Hence, $\mathcal{L}^{\uparrow}$ is closable. We still denote by $\mathcal{L}^{\uparrow}$ its extension, whose domain $\mathrm{dom}(\mathcal{L}^{\uparrow})$ contains at least $\mathcal{C}^{1,2}$. Consider the basis of $L^2(\mathbb{T}_2, \mathbb{C})$ given by

$$ e_{n,m}(x_1, x_2) = e^{2i\pi nx_1} e^{2i\pi mx_2}, \quad n, m \in \mathbb{Z}. $$

For $i = 1, 2$, we have in $L^2(\mathbb{T}_2, \mathbb{C})$,

$$ \phi^i = \sum_{n,m \in \mathbb{Z}} c_{n,m}^i e_{n,m}. $$
Furthermore,

$$ \phi_{22}^1 - \phi_{12}^1 = -4\pi^2 \sum_{n,m \in \mathbb{Z}} (c_{n,m}^1 m^2 - c_{n,m}^2 mn) e_{n,m}, $$

$$ \phi_{11}^1 - \phi_{12}^1 = -4\pi^2 \sum_{n,m \in \mathbb{Z}} (c_{n,m}^2 n^2 - c_{n,m}^1 mn) e_{n,m}. $$

Thus solving $\mathcal{L}^{\uparrow}\phi = -4\pi^2\lambda\phi$ amounts to find the $c_{n,m}^i$'s such that

$$ \begin{aligned} c_{n,m}^1(m^2 - \lambda) - c_{n,m}^2 mn &= 0, \\ -c_{n,m}^1 mn + c_{n,m}^2(n^2 - \lambda) &= 0. \end{aligned} $$

For $\lambda$ negative irrational, this system admits the null form as unique solution, hence for such a $\lambda$, $\mathcal{L}^{\uparrow} - 4\pi^2\lambda\text{Id}$ is one-to-one and the third condition is satisfied.

According to [17, Th.2.12 P.16], the operator $\mathcal{L}^{\uparrow}$ hence generates a strongly continuous contraction semi-group on $\mathcal{C}^{1,3}$. Moreover, by [17, Th.4.1 P.182], uniqueness holds for the martingale problem associated with $\mathcal{L}^{\uparrow}$. ■

## 4.4 Cycle random walk on the torus

### 4.4.1 Rescaled random walk on cycles

Consider the random walk $X^n$ with generator (37). It is a continuous-time jump process with values in $C_1^n \subset \mathcal{C}_{1,4}$ (the latter space does not depend on $n$), which we recall is a Polish space. The set $\mathcal{D}(\mathbb{R}_+, \mathcal{C}_{1,4})$ is embedded with the Skorokhod distance and is itself Polish (e.g. [28]). To prove the convergence, we will use the explicit expression of $X^n$ as the solution of a stochastic differential equation driven by Poisson point measures. Let $N_n(ds, d\theta, d\tau)$ be a Poisson point measure on $\mathbb{R}_+ \times \mathbb{R}_+ \times S_2^n$ with intensity measure $ds \otimes d\theta \otimes n(d\tau)$ where $n(d\tau)$ is the counting measure on $S_2^n$.

$$ X_t^n = X_0^n - \int_0^t \int_{\mathbb{R}_+} \int_{S_2^n} \partial_{2\tau} \mathbf{1}_{\{\theta \le w(X_{s-}^n, \partial_{2\tau})\}} N_n(ds, d\theta, d\tau). \quad (50) $$

In the sequel, we will study the convergence of the sequence of accelerated processes:

$$ X_t^{(n)} = X_{\epsilon_n^{-2}t}^n. \quad (51) $$

<figure><img src="image_9.png" /><figcaption>Figure 7: Simulations of the cycle-valued random walk on the simplicial complex composed of all the triangles of $\mathcal{T}_n$. Simulation by Paul Melotti.</figcaption></figure>
Furthermore,

$$ \phi_{22}^1 - \phi_{12}^1 = -4\pi^2 \sum_{n,m \in \mathbb{Z}} (c_{n,m}^1 m^2 - c_{n,m}^2 mn) e_{n,m}, $$

$$ \phi_{11}^1 - \phi_{12}^1 = -4\pi^2 \sum_{n,m \in \mathbb{Z}} (c_{n,m}^2 n^2 - c_{n,m}^1 mn) e_{n,m}. $$

Thus solving $\mathcal{L}^\uparrow \phi = -4\pi^2 \lambda \phi$ amounts to find the $c_{n,m}^i$'s such that

$$ \begin{aligned} c_{n,m}^1(m^2 - \lambda) - c_{n,m}^2 mn &= 0, \\ -c_{n,m}^1 mn + c_{n,m}^2(n^2 - \lambda) &= 0. \end{aligned} $$

For $\lambda$ negative irrational, this system admits the null form as unique solution, hence for such a $\lambda$, $\mathcal{L}^\uparrow - 4\pi^2 \lambda \mathrm{Id}$ is one-to-one and the third condition is satisfied.

According to [17, Th.2.12 P.16], the operator $\mathcal{L}^\uparrow$ hence generates a strongly continuous contraction semi-group on $\mathfrak{C}^{1,3}$. Moreover, by [17, Th.4.1 P.182], uniqueness holds for the martingale problem associated with $\mathcal{L}^\uparrow$. ■

## 4.4 Cycle random walk on the torus

### 4.4.1 Rescaled random walk on cycles

Consider the random walk $X^n$ with generator (37). It is a continuous-time jump process with values in $C_1^n \subset \mathfrak{C}_{1,4}$ (the latter space does not depend on $n$), which we recall is a Polish space. The set $D(\mathbb{R}_+, \mathfrak{C}_{1,4})$ is embedded with the Skorokhod distance and is itself Polish (e.g. [28]). To prove the convergence, we will use the explicit expression of $X^n$ as the solution of a stochastic differential equation driven by Poisson point measures. Let $N_n(ds, d\theta, d\tau)$ be a Poisson point measure on $\mathbb{R}_+ \times \mathbb{R}_+ \times S_2^n$ with intensity measure $ds \otimes d\theta \otimes n(d\tau)$ where $n(d\tau)$ is the counting measure on $S_2^n$.

$$ X_t^n = X_0^n - \int_0^t \int_{\mathbb{R}_+} \int_{S_2^n} \partial_{2\tau} \mathbf{1}_{\{\theta \le w(X_{s-}^n, \partial_{2\tau})\}} N_n(ds, d\theta, d\tau). \quad (50) $$

In the sequel, we will study the convergence of the sequence of accelerated processes:

$$ X_t^{(n)} = X_{\epsilon_n^{-2}t}^n. \quad (51) $$

<figure><img src="image_11.png" /><figcaption>Figure 7: Simulations of the cycle-valued random walk on the simplicial complex composed of all the triangles of $\mathcal{T}_n$. Simulation by Paul Melotti.</figcaption></figure>
### 4.4.2 Conjectures

As the random cycle at any time $t$ can have loops and use several times the same edge, one of the main difficulty in studying limit theorems for $(X_t^n)_{t \ge 0}$ lies in controlling its size. Indeed, for the usual convergence of random walks to the Brownian motion, we expect that the number of edges and the length of the curve tend to infinity.

We instead consider the flat norm of $X \in \mathfrak{C}_1$ (see [18, p.4]) that is defined as:

$$ \|X\|_F := \inf_{\Delta \in \mathfrak{C}_2} \|X - \partial_2 \Delta \|_{\mathfrak{C}_1} + \|\Delta\|_{\mathfrak{C}_2}. \qquad (52) $$

Roughly speaking, the infimum in the definition of the flat norm is over all the $\Delta \in \mathfrak{C}_2$ such that $\partial_2 \Delta$ is close to $X$ and that have a small area $\|\Delta\|_{\mathfrak{C}_2}$. In the following, we will need the following conjecture:

**Conjecture 33.** For all $T > 0$, we require the following control on the flat norm of our process:

$$ \lim_{n \to +\infty} \mathbb{E} \left( \sup_{t \le T} \| X_t^{(n)} \|_F \right) < +\infty \qquad (53) $$

Notice that with some modification of the support of the cycle-valued chain, we can show easily that the conjecture holds. If we introduce

$$ \sigma_0 = [(0,0), (2\epsilon_n, 0)] + [(2\epsilon_n, 0), (\epsilon_n, \sqrt{3}\epsilon_n)] + [(\epsilon_n, \sqrt{3}\epsilon_n), (0,0)], \text{ and } \tau_0 = [(0,0), (2\epsilon_n, 0), (\epsilon_n, \sqrt{3}\epsilon_n)], $$

then, for the simplicial complex $\tilde{C}_n := C_n \setminus \tau_0$, $H_1$ is of dimension 3 with the basis $\{\sigma_0, \sigma_1, \sigma_2\}$ and a chain starting in the homology class of $\sigma_0$ satisfies the condition (34). This ensures that $\Delta_t^n$ can be chosen as a combination of triangles with weights in $\{-1, 0, 1\}$ and also, it is possible to choose $\Delta_t^n$ such that $X_t^{(n)} - \partial_2 \Delta_t^n = X_0^{(n)}$. Therefore

$$ \|\Delta_t^n\|_{\mathfrak{C}_2} \le 2\sqrt{3}, $$

which is the area of $\mathbb{T}_2$. In this case, (53) is satisfied as soon as

$$ \mathbb{E}(\|X_0^{(n)}\|_{\mathfrak{C}_1}) < +\infty, \qquad (54) $$

a condition that depends only on the initial condition. However, rather than working on such a perforated torus we chose to stick with the random walk on the complete torus, since it is a more natural space, at the price of Conjecture 33. Another possible direction to avoid the Conjecture would be to consider chains with coefficients in $\mathbb{Z}/2\mathbb{Z}$.

As we mentioned earlier, we expect the length of our process to explode as $n$ goes to infinity. Still, our control on the flat norm of the process can be used to obtain an (exploding) bound on its length.

**Lemma 34.** For any 1-chain $\sigma$ of the simplicial complex $C_n$, we have

$$ \|\sigma\|_{\mathfrak{C}_1} \le \frac{2\sqrt{3}\|\sigma\|_F}{\epsilon_n}. $$

*Proof.* By the definition of the flat norm, there must exists a 1-cycle $\sigma_0 \in \mathfrak{C}_1$ and $\Delta \in \mathfrak{C}_2$ such that $\sigma = \sigma_0 + \partial_2 \Delta$ and

$$ \|\sigma\|_F = \|\sigma_0\|_{\mathfrak{C}_1} + \|\Delta\|_{\mathfrak{C}_2}. $$

Since $\sigma$ is a 1-chain of the simplicial complex $C^n$, $\sigma_0$ is also a 1-chain of $C^n$ and $\Delta$ a 2-chain of $C^n$. We thus have $\Delta = \sum_{\tau \in S_2^n} \lambda_\tau \tau$, where $\forall \tau \in S_2^n$, $\lambda_\tau \in \mathbb{R}$ and

$$ \|\Delta\|_{\mathfrak{C}_2} = \sqrt{3}\epsilon_n^2 \sum_{\tau \in S_2^n} |\lambda_\tau|. $$

Therefore,

$$ \|\partial_2 \Delta\|_{\mathfrak{C}_1} \le \sum_{\tau \in S_2^n} |\lambda_\tau| \|\partial_2 \tau\|_{\mathfrak{C}_1} = 6\epsilon_n \sum_{\tau \in S_2^n} |\lambda_\tau| = \frac{2\sqrt{3}\|\Delta\|_{\mathfrak{C}_2}}{\epsilon_n} $$
### 4.4.2 Conjectures

As the random cycle at any time $t$ can have loops and use several times the same edge, one of the main difficulty in studying limit theorems for $(X_t^n)_{t \ge 0}$ lies in controlling its size. Indeed, for the usual convergence of random walks to the Brownian motion, we expect that the number of edges and the length of the curve tend to infinity.

We instead consider the flat norm of $X \in \mathfrak{C}_1$ (see [18, p.4]) that is defined as:

$$ \|X\|_F := \inf_{\Delta \in \mathfrak{C}_2} \|X - \partial_2 \Delta \|_{\mathfrak{C}_1} + \|\Delta\|_{\mathfrak{C}_2}. \quad (52) $$

Roughly speaking, the infimum in the definition of the flat norm is over all the $\Delta \in \mathfrak{C}_2$ such that $\partial_2 \Delta$ is close to $X$ and that have a small area $\|\Delta\|_{\mathfrak{C}_2}$. In the following, we will need the following conjecture:

**Conjecture 33.** For all $T > 0$, we require the following control on the flat norm of our process:

$$ \lim_{n \to +\infty} \mathbb{E} \left( \sup_{t \le T} \| X_t^{(n)} \|_F \right) < +\infty \quad (53) $$

Notice that with some modification of the support of the cycle-valued chain, we can show easily that the conjecture holds. If we introduce

$$ \sigma_0 = [(0,0), (2\epsilon_n, 0)] + [(2\epsilon_n, 0), (\epsilon_n, \sqrt{3}\epsilon_n)] + [(\epsilon_n, \sqrt{3}\epsilon_n), (0,0)], \text{ and } \tau_0 = [(0,0), (2\epsilon_n, 0), (\epsilon_n, \sqrt{3}\epsilon_n)], $$

then, for the simplicial complex $\tilde{C}_n := C_n \setminus \tau_0$, $H_1$ is of dimension 3 with the basis $\{\sigma_0, \sigma_1, \sigma_2\}$ and a chain starting in the homology class of $\sigma_0$ satisfies the condition (34). This ensures that $\Delta_t^n$ can be chosen as a combination of triangles with weights in $\{-1, 0, 1\}$ and also, it is possible to choose $\Delta_t^n$ such that $X_t^{(n)} - \partial_2 \Delta_t^n = X_0^{(n)}$. Therefore

$$ \|\Delta_t^n\|_{\mathfrak{C}_2} \le 2\sqrt{3}, $$

which is the area of $\mathbb{T}_2$. In this case, (53) is satisfied as soon as

$$ \mathbb{E}(\|X_0^{(n)}\|_{\mathfrak{C}_1}) < +\infty, \quad (54) $$

a condition that depends only on the initial condition. However, rather than working on such a perforated torus we chose to stick with the random walk on the complete torus, since it is a more natural space, at the price of Conjecture 33. Another possible direction to avoid the Conjecture would be to consider chains with coefficients in $\mathbb{Z}/2\mathbb{Z}$.

As we mentioned earlier, we expect the length of our process to explode as $n$ goes to infinity. Still, our control on the flat norm of the process can be used to obtain an (exploding) bound on its length.

**Lemma 34.** For any 1-chain $\sigma$ of the simplicial complex $C_n$, we have

$$ \|\sigma\|_{\mathfrak{C}_1} \le \frac{2\sqrt{3}\|\sigma\|_F}{\epsilon_n}. $$

*Proof.* By the definition of the flat norm, there must exists a 1-cycle $\sigma_0 \in \mathfrak{C}_1$ and $\Delta \in \mathfrak{C}_2$ such that $\sigma = \sigma_0 + \partial_2 \Delta$ and

$$ \|\sigma\|_F = \|\sigma_0\|_{\mathfrak{C}_1} + \|\Delta\|_{\mathfrak{C}_2}. $$

Since $\sigma$ is a 1-chain of the simplicial complex $C^n$, $\sigma_0$ is also a 1-chain of $C^n$ and $\Delta$ a 2-chain of $C^n$. We thus have $\Delta = \sum_{\tau \in S_2^n} \lambda_\tau \tau$, where $\forall \tau \in S_2^n$, $\lambda_\tau \in \mathbb{R}$ and

$$ \|\Delta\|_{\mathfrak{C}_2} = \sqrt{3}\epsilon_n^2 \sum_{\tau \in S_2^n} |\lambda_\tau|. $$

Therefore,

$$ \|\partial_2 \Delta\|_{\mathfrak{C}_1} \le \sum_{\tau \in S_2^n} |\lambda_\tau| \|\partial_2 \tau\|_{\mathfrak{C}_1} = 6\epsilon_n \sum_{\tau \in S_2^n} |\lambda_\tau| = \frac{2\sqrt{3}\|\Delta\|_{\mathfrak{C}_2}}{\epsilon_n} $$
and

$$ \|\sigma\|_{\mathcal{C}_1} \le \|\sigma_0\|_{\mathcal{C}_1} + \|\partial_2 \Delta\|_{\mathcal{C}_1} \le \frac{2\sqrt{3}(\|\sigma_0\|_{\mathcal{C}_1} + \|\Delta\|_{\mathcal{C}_2})}{\epsilon_n}, $$

which concludes the proof.

<figure><img src="image_10.png" /></figure>

As a consequence of this result, Conjecture 33 implies that

$$ \lim_{n \to +\infty} \mathbb{E} \left( \sup_{t \le T} \epsilon_n \|X_t^{(n)}\|_{\mathcal{C}_1} \right) < +\infty. \quad (55) $$

We can also control the norms $\|\cdot\|_{\mathcal{C}_{1,4}}$ under the Conjecture 33 and obtain non-exploding bounds.

**Proposition 35.** Under the Conjecture 33, we have for any $T > 0$

$$ \sup_{n \in \mathbb{N}} \mathbb{E} \left( \sup_{t \le T} \| X_t^{(n)} \|_{\mathcal{C}_{1,4}} \right) < +\infty. \quad (56) $$

*Proof.* Let $n \in \mathbb{N}$ and let $\Delta^n \in \mathcal{C}_2^n$ be a chain of triangles. We have:

$$ \begin{aligned} \|X_t^{(n)}\|_{\mathcal{C}_{1,4}} &= \sup_{f \in \mathcal{C}_{1,4}^{(n)}} \frac{\langle X_t^{(n)}, f \rangle_{\mathcal{C}_{1,4}, \mathcal{C}_{1,4}^{(n)}}}{\|f\|_{\mathcal{C}_{1,4}^{(n)}}} \\ &\le \sup_{f \in \mathcal{C}_{1,4}^{(n)}} \frac{\langle X_t^{(n)} - \partial_2 \Delta^n, f \rangle_{\mathcal{C}_{1,4}, \mathcal{C}_{1,4}^{(n)}}}{\|f\|_{\mathcal{C}_{1,4}^{(n)}}} + \sup_{f \in \mathcal{C}_{1,4}^{(n)}} \frac{\langle \Delta^n, \partial_2^* f \rangle_{\mathcal{C}_{2,3}, \mathcal{C}_{2,3}^{(n)}}}{\|f\|_{\mathcal{C}_{1,4}^{(n)}}} \\ &\le \|X_t^{(n)} - \partial_2 \Delta^n\|_{\mathcal{C}_{1,4}} + \|\Delta^n\|_{\mathcal{C}_{2,3}} \end{aligned} \quad (57) $$

by using that

$$ \|\partial_2^* f\|_{\mathcal{C}_{2,3}} \le \|f\|_{\mathcal{C}_{1,4}}, \quad (58) $$

by (46). Taking the infimum in the right hand side with respect to $\Delta^n$, we obtain the flat norm $\|X_t^{(n)}\|_F$ and we conclude with the Conjecture 33. ■

### 4.4.3 Main result

Let us first introduce a notation. For a sequence $\sigma \in \mathcal{C}_1$, let $\tilde{\sigma}$ be the measure defined for any $\psi \in \mathcal{C}(\mathbb{T}_2, \mathbb{R})$ and any sequence $(\sigma_n)_{n \ge 1}$ converging to $\sigma$ such that $\limsup_{n \to +\infty} \epsilon_n \|\sigma_n\|_{\mathcal{C}_1} < +\infty$, by:

$$ \langle \tilde{\sigma}, \psi \rangle = \lim_{n \to +\infty} \epsilon_n \langle \sigma_n, \psi \rangle. \quad (59) $$

This measure is a kind of uniform measure along the cycle $\sigma$ (whose length is possible infinite).

The main result of this section is that:

**Theorem 36.** Assume that the Conjecture 33 holds and that the initial conditions $(X_0^{(n)})_{n \ge 1}$ converge in probability and in $\mathcal{C}_{1,4}$ to a limit $X_0$ and are such that:

$$ \sup_{n \in \mathbb{N}} \mathbb{E} \left( \| X_0^{(n)} \|_{\mathcal{C}_{1,4}} \right) < +\infty. \quad (60) $$

The limiting values $(X_t)_{t \in \mathbb{R}_+}$ in $\mathbb{D}(\mathbb{R}_+, \mathcal{C}_{1,4})$ of the sequence of random walks $(X^{(n)})_{n \ge 1}$ all solve the martingale problem in $\mathcal{C}(\mathbb{R}_+, \mathcal{C}_{1,4})$ associated with the generator $\mathfrak{A}$, defined for functions $F_\phi : \sigma \in \mathcal{C}_{1,4} \mapsto F(\langle \sigma, \phi \rangle_{\mathcal{C}_{1,4}, \mathcal{C}_{1,4}^{(n)}})$ by:

$$ \mathfrak{A}F_{\phi}(\sigma) = F'\left(\int_{\sigma} \phi\right) \int_{\sigma} \mathcal{L}^{\uparrow}\phi + 3F''\left(\int_{\sigma} \phi\right) \langle \tilde{\sigma}, (\ast d\phi)^2 \rangle, \quad (61) $$

More precisely, the process:

$$ M_t := F_{\phi}(X_t) - F_{\phi}(X_0) - \int_0^t \mathfrak{A}F_{\phi}(X_s)ds, \quad (62) $$
and

$$ \|\sigma\|_{\mathcal{C}_1} \le \|\sigma_0\|_{\mathcal{C}_1} + \|\partial_2 \Delta\|_{\mathcal{C}_1} \le \frac{2\sqrt{3}(\|\sigma_0\|_{\mathcal{C}_1} + \|\Delta\|_{\mathcal{C}_2})}{\epsilon_n}, $$

which concludes the proof.

<figure><img src="image_12.png" /></figure>

As a consequence of this result, Conjecture 33 implies that

$$ \lim_{n \to +\infty} \mathbb{E} \left( \sup_{t \le T} \epsilon_n \| X_t^{(n)} \|_{\mathcal{C}_1} \right) < +\infty. \quad (55) $$

We can also control the norms $\|\cdot\|_{\mathcal{C}_{1,4}}$ under the Conjecture 33 and obtain non-exploding bounds.

**Proposition 35.** Under the Conjecture 33, we have for any $T > 0$

$$ \sup_{n \in \mathbb{N}} \mathbb{E} \left( \sup_{t \le T} \| X_t^{(n)} \|_{\mathcal{C}_{1,4}} \right) < +\infty. \quad (56) $$

*Proof.* Let $n \in \mathbb{N}$ and let $\Delta^n \in \mathcal{C}_2^n$ be a chain of triangles. We have:

$$ \begin{aligned} \|X_t^{(n)}\|_{\mathcal{C}_{1,4}} &= \sup_{f \in \mathcal{C}_{1,4}^{(n)}} \frac{\langle X_t^{(n)}, f \rangle_{\mathcal{C}_{1,4}, \mathcal{C}_{1,4}^{(n)}}}{\|f\|_{\mathcal{C}_{1,4}^{(n)}}} \\ &\le \sup_{f \in \mathcal{C}_{1,4}^{(n)}} \frac{\langle X_t^{(n)} - \partial_2 \Delta^n, f \rangle_{\mathcal{C}_{1,4}, \mathcal{C}_{1,4}^{(n)}}}{\|f\|_{\mathcal{C}_{1,4}^{(n)}}} + \sup_{f \in \mathcal{C}_{1,4}^{(n)}} \frac{\langle \Delta^n, \partial_2^* f \rangle_{\mathcal{C}_{2,3}, \mathcal{C}_{2,3}^{(n)}}}{\|f\|_{\mathcal{C}_{1,4}^{(n)}}} \\ &\le \|X_t^{(n)}\|_{\mathcal{C}_{1,4}} - \partial_2 \Delta^n \|_{\mathcal{C}_{1,4}} + \|\Delta^n\|_{\mathcal{C}_{2,3}} \end{aligned} \quad (57) $$

by using that

$$ \|\partial_2^* f\|_{\mathcal{C}_{2,3}^{(n)}} \le \|f\|_{\mathcal{C}_{1,4}^{(n)}}, \quad (58) $$

by (46). Taking the infinimum in the right hand side with respect to $\Delta^n$, we obtain the flat norm $\|X_t^{(n)}\|_F$ and we conclude with the Conjecture 33. ■

<figure><img src="image_13.png" /></figure>

### 4.4.3 Main result

Let us first introduce a notation. For a sequence $\sigma \in \mathcal{C}_1$, let $\tilde{\sigma}$ be the measure defined for any $\psi \in \mathcal{C}(\mathbb{T}_2, \mathbb{R})$ and any sequence $(\sigma_n)_{n \ge 1}$ converging to $\sigma$ such that $\limsup_{n \to +\infty} \epsilon_n \|\sigma_n\|_{\mathcal{C}_1} < +\infty$, by:

$$ \langle \tilde{\sigma}, \psi \rangle = \lim_{n \to +\infty} \epsilon_n \langle \sigma_n, \psi \rangle. \quad (59) $$

This measure is a kind of uniform measure along the cycle $\sigma$ (whose length is possible infinite).

The main result of this section is that:

**Theorem 36.** Assume that the Conjecture 33 holds and that the initial conditions $(X_0^{(n)})_{n \ge 1}$ converge in probability and in $\mathcal{C}_{1,4}$ to a limit $X_0$ and are such that:

$$ \sup_{n \in \mathbb{N}} \mathbb{E} \left( \| X_0^{(n)} \|_{\mathcal{C}_{1,4}} \right) < +\infty. \quad (60) $$

The limiting values $(X_t)_{t \in \mathbb{R}_+}$ in $\mathbb{D}(\mathbb{R}_+, \mathcal{C}_{1,4})$ of the sequence of random walks $(X^{(n)})_{n \ge 1}$ all solve the martingale problem in $\mathcal{C}(\mathbb{R}_+, \mathcal{C}_{1,4})$ associated with the generator $\mathfrak{A}$, defined for functions $F_\phi : \sigma \in \mathcal{C}_{1,4} \mapsto F(\langle \sigma, \phi \rangle_{\mathcal{C}_{1,4}, \mathcal{C}_{1,4}^{(n)}})$ by:

$$ \mathfrak{A}F_{\phi}(\sigma) = F'\left(\int_{\sigma} \phi\right) \int_{\sigma} \mathcal{L}^{\uparrow}\phi + 3F''\left(\int_{\sigma} \phi\right) \langle \tilde{\sigma}, (\ast d\phi)^2 \rangle, \quad (61) $$

More precisely, the process:

$$ M_t := F_{\phi}(X_t) - F_{\phi}(X_0) - \int_0^t \mathfrak{A}F_{\phi}(X_s)ds, \quad (62) $$
is a square integrable real-valued martingale with predictable quadratic variation process:

$$ \begin{aligned} \langle M \rangle_t &= \lim_{n \to +\infty} \epsilon_n^{-2} \int_0^t \sum_{\tau \in S_2^n} \langle \partial_2 \tau, \phi \rangle^2 (F'(\langle X_u^{(n)}, \phi \rangle)^2 w(X_u^{(n)}, \partial_2 \tau) du \\ &= \int_0^t F'_\phi(X_u)^2 \left( \int_{\mathbb{T}_2} \left( \frac{\partial \phi^2}{\partial x_1} - \frac{\partial \phi^1}{\partial x_2} \right)^2 \tilde{X}_u(\, dx_1, \, dx_2) \right) du, \end{aligned} \quad (63) $$

where $\tilde{X}$ is defined in (59).

Notice that if $F$ is linear (for instance $F(x) = x$), the second term in (61) disappears and we recover $\mathcal{L}^\dagger$ in the first term.

The end of this Section 4 is devoted to the proof of Theorem 36. We first show that the generators of these random walks converge in $\mathbb{C}^{1,4}$ to the closure of $(\mathfrak{A}, \mathbb{C}_{1,4})$. Then, we prove the tightness of the sequence of distributions of the $X^{(n)}$'s on $\mathbb{C}_{1,4}$. The uniqueness of the limiting martingale problem solved by the limiting values remains open.

## 4.5 Proof of Theorem 36

The proof of Theorem 36 is divided into several steps. We first start with some preliminary computation in Section 4.5.1. In Section 4.5.2, we show that the generators $\mathcal{A}_n$ restricted to functions $F_\phi(\sigma) = \langle \sigma, \phi \rangle$ converge to $\mathcal{L}^\dagger$. The difficulty here is to control the error carefully so that we can show tightness of the sequence of distributions of $(X^{(n)})$ in Section 4.5.3 and convergence of the generators $\mathcal{A}_n$ to $\mathfrak{A}$ in Section 4.5.4.

### 4.5.1 Preliminary estimates

Let us first establish some estimates that will be needed.

**Lemma 37.** Let $\phi \in \mathbb{C}^{1,4}$, with $\phi(x) = \phi^1(x)dx_1 + \phi^2(x)dx_2$, and let $\tau \in S_2^n$. Without loss of generality, we can assume (by a change of coordinates) that $\tau = [021]$ (see Fig. 6). Then,

$$ \langle \partial_2 \tau, \phi \rangle_{\mathcal{E}_{1,4}, \mathcal{E}^{1,4}} = \sqrt{3} \epsilon_n^2 (\phi_1^2(0,0) - \phi_2^1(0,0)) + \sqrt{3} \epsilon_n^3 (\phi_{11}^2(0,0) - \phi_{12}^1(0,0)) \\ + \epsilon_n^3 (\phi_{12}^2(0,0) - \phi_{22}^1(0,0)) + O(\epsilon_n^4) \quad (64) $$

$$ = \frac{\sqrt{3}}{2} \epsilon_n \int_0^{2\epsilon_n} (\phi_1^2(x_1, 0) - \phi_2^1(x_1, 0)) \, dx_1 + O(\epsilon_n^3). \quad (65) $$

*Proof.* Using the Stoke's formula (e.g. [5, Th.6.2.1]):

$$ \langle \partial_2 \tau, \phi \rangle = \int_{\tau} d\phi = \int_{\tau} (\phi_1^2 - \phi_2^1) dx_1 \wedge dx_2, $$

where the exterior derivative of $\phi$, $d\phi$, has been computed in (46). We then obtain the result using Lemma 46 in Appendix with $H(x_1, x_2) = \phi_1^2(x_1, x_2) - \phi_2^1(x_1, x_2)$, that is of class $\mathcal{C}^2$ under our assumption. ■

**Lemma 38.** Let $H: \mathbb{T}_2 \to \mathbb{R}$ be $\mathcal{C}^3$. Let us consider the edge $[0, 1]$ and the two adjacent triangles $[021]$ and $[031]$ where the vertices 0, 1, 2 and 3 have coordinate $(0,0)$, $(\epsilon_n, \sqrt{3}\epsilon_n)$ and $(2\epsilon_n, 0)$ (see Figure 6). Then, as $\epsilon_n$ goes to 0,

$$ \epsilon_n^{-2} \int_{\tau^+} H(x_1, x_2) \, dx_2 \, dx_1 + \epsilon_n^{-2} \int_{\tau^-} H(x_1, x_2) \, dx_2 \, dx_1 = \int_0^{2\epsilon_n} 3c(x_1, 0)H_2(x_1, 0) \, dx_1 + O(\epsilon_n^3) \quad (66) $$

where $c(x) = \min_{x' \in V_n} \left( \frac{\|x-x'\|}{\epsilon_n} \right)^2$.
is a square integrable real-valued martingale with predictable quadratic variation process:

$$ \begin{aligned} \langle M \rangle_t &= \lim_{n \to +\infty} \epsilon_n^{-2} \int_0^t \sum_{\tau \in S_2^n} \langle \partial_2 \tau, \phi \rangle^2 (F'(\langle X_u^{(n)}, \phi \rangle)^2 w(X_u^{(n)}, \partial_2 \tau) du \\ &= \int_0^t F'_\phi(X_u)^2 \left( \int_{\mathbb{T}_2} \left( \frac{\partial \phi^2}{\partial x_1} - \frac{\partial \phi^1}{\partial x_2} \right)^2 \tilde{X}_u(\, dx_1, \, dx_2) \right) du, \end{aligned} \quad (63) $$

where $\tilde{X}$ is defined in (59).

Notice that if $F$ is linear (for instance $F(x) = x$), the second term in (61) disappears and we recover $\mathcal{L}^\dagger$ in the first term.

The end of this Section 4 is devoted to the proof of Theorem 36. We first show that the generators of these random walks converge in $\mathbb{C}^{1,4}$ to the closure of $(\mathfrak{A}, \mathbb{C}_{1,4})$. Then, we prove the tightness of the sequence of distributions of the $X^{(n)}$'s on $\mathbb{C}_{1,4}$. The uniqueness of the limiting martingale problem solved by the limiting values remains open.

## 4.5 Proof of Theorem 36

The proof of Theorem 36 is divided into several steps. We first start with some preliminary computation in Section 4.5.1. In Section 4.5.2, we show that the generators $\mathcal{A}_n$ restricted to functions $F_\phi(\sigma) = \langle \sigma, \phi \rangle$ converge to $\mathcal{L}^\dagger$. The difficulty here is to control the error carefully so that we can show tightness of the sequence of distributions of $(X^{(n)})$ in Section 4.5.3 and convergence of the generators $\mathcal{A}_n$ to $\mathfrak{A}$ in Section 4.5.4.

### 4.5.1 Preliminary estimates

Let us first establish some estimates that will be needed.

**Lemma 37.** Let $\phi \in \mathbb{C}^{1,4}$, with $\phi(x) = \phi^1(x)dx_1 + \phi^2(x)dx_2$, and let $\tau \in S_2^n$. Without loss of generality, we can assume (by a change of coordinates) that $\tau = [021]$ (see Fig. 6). Then,

$$ \langle \partial_2 \tau, \phi \rangle_{\mathcal{L}_{1,4}, \mathcal{L}^{1,4}} = \sqrt{3} \epsilon_n^2 (\phi_1^2(0,0) - \phi_2^1(0,0)) + \sqrt{3} \epsilon_n^3 (\phi_{11}^2(0,0) - \phi_{12}^1(0,0)) \\ + \epsilon_n^3 (\phi_{12}^2(0,0) - \phi_{22}^1(0,0)) + O(\epsilon_n^4) \quad (64) $$

$$ = \frac{\sqrt{3}}{2} \epsilon_n \int_0^{2\epsilon_n} (\phi_1^2(x_1, 0) - \phi_2^1(x_1, 0)) dx_1 + O(\epsilon_n^3). \quad (65) $$

*Proof.* Using the Stoke's formula (e.g. [5, Th.6.2.1]):

$$ \langle \partial_2 \tau, \phi \rangle = \int_{\tau} d\phi = \int_{\tau} (\phi_1^2 - \phi_2^1) dx_1 \wedge dx_2, $$

where the exterior derivative of $\phi$, $d\phi$, has been computed in (46). We then obtain the result using Lemma 46 in Appendix with $H(x_1, x_2) = \phi_1^2(x_1, x_2) - \phi_2^1(x_1, x_2)$, that is of class $\mathcal{C}^2$ under our assumption. ■

**Lemma 38.** Let $H: \mathbb{T}_2 \to \mathbb{R}$ be $\mathcal{C}^3$. Let us consider the edge $[0, 1]$ and the two adjacent triangles $[021]$ and $[031]$ where the vertices 0, 1, 2 and 3 have coordinate $(0,0)$, $(\epsilon_n, \sqrt{3}\epsilon_n)$ and $(2\epsilon_n, 0)$ (see Figure 6). Then, as $\epsilon_n$ goes to 0,

$$ \epsilon_n^{-2} \int_{\tau^+} H(x_1, x_2) \, dx_2 \, dx_1 + \epsilon_n^{-2} \int_{\tau^-} H(x_1, x_2) \, dx_2 \, dx_1 = \int_0^{2\epsilon_n} 3c(x_1, 0)H_2(x_1, 0) \, dx_1 + O(\epsilon_n^3) \quad (66) $$

where $c(x) = \min_{x' \in V_n} \left( \frac{\|x-x'\|}{\epsilon_n} \right)^2$.
Proof. We have:

$$
\begin{align*}
A := \epsilon_n^{-2} \int_{\tau^+} H(x_1, x_2) \, dx_2 \, dx_1 + \epsilon_n^{-2} \int_{\tau^-} H(x_1, x_2) \, dx_2 \, dx_1 \\
&= \epsilon_n^{-2} \int_0^{\epsilon_n} \int_0^{\sqrt{3}x_1} (H(x_1, x_2) - H(x_1, -x_2)) \, dx_2 \, dx_1 \\
&\quad + \epsilon_n^{-2} \int_{\epsilon_n}^{2\epsilon_n} \int_0^{\sqrt{3}(2\epsilon_n - x_1)} (H(x_1, x_2) - H(x_1, -x_2)) \, dx_2 \, dx_1.
\end{align*}
$$

For a given point $(x_1, x_2)$, a Taylor expansion gives that, for some $\xi_{x_1,x_2} \in (0, x_2)$,

$$
H(x_1, x_2) = H(x_1, 0) + x_2 H_2(x_1, 0) + \frac{x_2^2}{2} H_2(x_1, 0) + \frac{x_2^3}{6} H_2(x_1, \xi_{x_1, x_2}), \quad (67)
$$

$$
H(x_1, -x_2) = H(x_1, 0) + x_2 H_2(x_1, 0) + \frac{x_2^2}{2} H_2(x_1, 0) + \frac{x_2^3}{6} H_2(x_1, \xi_{x_1, -x_2}). \quad (68)
$$

Thus:

$$
\begin{align*}
A ={}& \epsilon_n^{-2} \int_0^{\epsilon_n} \int_0^{\sqrt{3}x_1} \left( 2x_2 H_2(x_1, 0) + \frac{x_2^3}{6} \left( H_{222}(x_1, \xi_{x_1,x_2}) + H_{222}(x_1, \xi_{x_1,-x_2}) \right) \right) dx_2 \, dx_1 \\
& + \epsilon_n^{-2} \int_{\epsilon_n}^{2\epsilon_n} \int_0^{\sqrt{3}(2\epsilon_n - x_1)} \left( 2x_2 H_2(x_1, 0) + \frac{x_2^3}{6} \left( H_{222}(x_1, \xi_{x_1,x_2}) + H_{222}(x_1, \xi_{x_1,-x_2}) \right) \right) dx_2 \, dx_1 \\
={}& \int_0^{\epsilon_n} 3 \left( \frac{x_1}{\epsilon_n} \right)^2 H_2(x_1, 0) \, dx_1 + \int_{\epsilon_n}^{2\epsilon_n} 3 \left( \frac{2\epsilon_n - x_1}{\epsilon_n} \right)^2 H_2(x_1, 0) \, dx_1 + O(\epsilon_n^3).
\end{align*}
$$

<figure><img src="image_11.png" /></figure>

4.5.2 Convergence of the generators for linear maps

From the above lemma, we can show that the generators $\mathcal{A}_n$ converge to $\mathcal{L}^\uparrow$. The limiting generator is defined for twice differentiable 1-forms, but for establishing convergences, we will require some more regularity and will consider $\mathcal{C}^{1,4}$.

**Proposition 39.** For any 1-cycle $\sigma$ and for a function $\phi \in \mathcal{C}^{1,4}(\mathbb{T}_2)$, we have

$$
\mathcal{A}_n(\langle\sigma, \phi\rangle) = \int_{\sigma} \mathcal{L}^{\uparrow}\phi + O(\epsilon_n \|\sigma\|_F),
$$

where $\|\cdot\|_F$ is the flat norm defined in (52).

Proof. Consider a cycle $\sigma$, we have

$$
\begin{align}
\mathcal{A}_n(\langle\sigma, \phi\rangle) &= \sum_{e \in \sigma} \epsilon_n^{-2} \left( \langle e - \partial_2 \tau^+, \phi \rangle_{\mathcal{C}_{1,3}, \mathcal{C}^{1,3}} - \langle e, \phi \rangle_{\mathcal{C}_{1,3}, \mathcal{C}^{1,3}} \right) + \epsilon_n^{-2} \left( \langle e - \partial_2 \tau^-, \phi \rangle_{\mathcal{C}_{1,3}, \mathcal{C}^{1,3}} - \langle e, \phi \rangle_{\mathcal{C}_{1,3}, \mathcal{C}^{1,3}} \right) \nonumber \\
&= \sum_{e \in \sigma} -\epsilon_n^{-2} \left( \langle \partial_2 \tau^+, \phi \rangle_{\mathcal{C}_{1,3}, \mathcal{C}^{1,3}} + \langle \partial_2 \tau^-, \phi \rangle_{\mathcal{C}_{1,3}, \mathcal{C}^{1,3}} \right) = -\sum_{e \in \sigma} \epsilon_n^{-2} \left( \int_{\tau^+} d\phi + \int_{\tau^-} d\phi \right), \tag{69}
\end{align}
$$

where for each edge $e \in \sigma$, we denote by $\tau^+$ and $\tau^-$ the adjacent cofaces, which have different orientations.
Then, in view of (46), taking $H(x_1, x_2) = \phi_1^2(x_1, x_2) - \phi_2^1(x_1, x_2)$ (which is of class $C^3$ by our assumptions)
in Lemma 38, we obtain

$$
\mathcal{A}_n(\langle\sigma, \phi\rangle) = \int_{\sigma} 3c \mathcal{L}^{\uparrow}\phi + O(\epsilon_n^2 \|\sigma\|_{\mathcal{C}^1})
$$

and, by Lemma 34,

$$
\mathcal{A}_n(\langle\sigma, \phi\rangle) = \int_{\sigma} 3c \mathcal{L}^{\uparrow}\phi + O(\epsilon_n \|\sigma\|_F).
$$
Proof. We have:

$$
\begin{align*}
A := \epsilon_n^{-2} \int_{\tau^+} H(x_1, x_2) \, dx_2 \, dx_1 + \epsilon_n^{-2} \int_{\tau^-} H(x_1, x_2) \, dx_2 \, dx_1 \\
&= \epsilon_n^{-2} \int_0^{\epsilon_n} \int_0^{\sqrt{3}x_1} (H(x_1, x_2) - H(x_1, -x_2)) \, dx_2 \, dx_1 \\
&\quad + \epsilon_n^{-2} \int_{\epsilon_n}^{2\epsilon_n} \int_0^{\sqrt{3}(2\epsilon_n - x_1)} (H(x_1, x_2) - H(x_1, -x_2)) \, dx_2 \, dx_1.
\end{align*}
$$

For a given point $(x_1, x_2)$, a Taylor expansion gives that, for some $\xi_{x_1,x_2} \in (0, x_2)$,

$$
H(x_1, x_2) = H(x_1, 0) + x_2 H_2(x_1, 0) + \frac{x_2^2}{2} H_2(x_1, 0) + \frac{x_2^3}{6} H_2(x_1, \xi_{x_1, x_2}), \quad (67)
$$

$$
H(x_1, -x_2) = H(x_1, 0) + x_2 H_2(x_1, 0) + \frac{x_2^2}{2} H_2(x_1, 0) + \frac{x_2^3}{6} H_2(x_1, \xi_{x_1, -x_2}). \quad (68)
$$

Thus:

$$
\begin{align*}
A ={}& \epsilon_n^{-2} \int_0^{\epsilon_n} \int_0^{\sqrt{3}x_1} \left( 2x_2 H_2(x_1, 0) + \frac{x_2^3}{6} \left( H_{222}(x_1, \xi_{x_1,x_2}) + H_{222}(x_1, \xi_{x_1,-x_2}) \right) \right) dx_2 \, dx_1 \\
& + \epsilon_n^{-2} \int_{\epsilon_n}^{2\epsilon_n} \int_0^{\sqrt{3}(2\epsilon_n - x_1)} \left( 2x_2 H_2(x_1, 0) + \frac{x_2^3}{6} \left( H_{222}(x_1, \xi_{x_1,x_2}) + H_{222}(x_1, \xi_{x_1,-x_2}) \right) \right) dx_2 \, dx_1 \\
={}& \int_0^{\epsilon_n} 3 \left( \frac{x_1}{\epsilon_n} \right)^2 H_2(x_1, 0) \, dx_1 + \int_{\epsilon_n}^{2\epsilon_n} 3 \left( \frac{2\epsilon_n - x_1}{\epsilon_n} \right)^2 H_2(x_1, 0) \, dx_1 + O(\epsilon_n^3).
\end{align*}
$$

<figure><img src="image_14.png" /></figure>

4.5.2 Convergence of the generators for linear maps

From the above lemma, we can show that the generators $\mathcal{A}_n$ converge to $\mathcal{L}^\uparrow$. The limiting generator is defined for twice differentiable 1-forms, but for establishing convergences, we will require some more regularity and will consider $\mathcal{C}^{1,4}$.

**Proposition 39.** For any 1-cycle $\sigma$ and for a function $\phi \in \mathcal{C}^{1,4}(\mathbb{T}_2)$, we have

$$
\mathcal{A}_n(\langle\sigma, \phi\rangle) = \int_{\sigma} \mathcal{L}^{\uparrow}\phi + O(\epsilon_n \|\sigma\|_F),
$$

where $\|\cdot\|_F$ is the flat norm defined in (52).

Proof. Consider a cycle $\sigma$, we have

$$
\begin{align}
\mathcal{A}_n(\langle\sigma, \phi\rangle) &= \sum_{e \in \sigma} \epsilon_n^{-2} \left( \langle e - \partial_2 \tau^+, \phi \rangle_{\mathcal{C}_{1,3}, \mathcal{C}^{1,3}} - \langle e, \phi \rangle_{\mathcal{C}_{1,3}, \mathcal{C}^{1,3}} \right) + \epsilon_n^{-2} \left( \langle e - \partial_2 \tau^-, \phi \rangle_{\mathcal{C}_{1,3}, \mathcal{C}^{1,3}} - \langle e, \phi \rangle_{\mathcal{C}_{1,3}, \mathcal{C}^{1,3}} \right) \nonumber \\
&= \sum_{e \in \sigma} -\epsilon_n^{-2} \left( \langle \partial_2 \tau^+, \phi \rangle_{\mathcal{C}_{1,3}, \mathcal{C}^{1,3}} + \langle \partial_2 \tau^-, \phi \rangle_{\mathcal{C}_{1,3}, \mathcal{C}^{1,3}} \right) = -\sum_{e \in \sigma} \epsilon_n^{-2} \left( \int_{\tau^+} d\phi + \int_{\tau^-} d\phi \right), \tag{69}
\end{align}
$$

where for each edge $e \in \sigma$, we denote by $\tau^+$ and $\tau^-$ the adjacent cofaces, which have different orientations.
Then, in view of (46), taking $H(x_1, x_2) = \phi_1^2(x_1, x_2) - \phi_2^1(x_1, x_2)$ (which is of class $\mathcal{C}^3$ by our assumptions)
in Lemma 38, we obtain

$$
\mathcal{A}_n(\langle\sigma, \phi\rangle) = \int_{\sigma} 3c \mathcal{L}^{\uparrow}\phi + O(\epsilon_n^2 \|\sigma\|_{\mathcal{C}^1})
$$

and, by Lemma 34,

$$
\mathcal{A}_n(\langle\sigma, \phi\rangle) = \int_{\sigma} 3c \mathcal{L}^{\uparrow}\phi + O(\epsilon_n \|\sigma\|_F).
$$
Now, let $\sigma_0$ and $\Delta$ be a cycle and a 2-chain (in the simplicial complex $C_n$) such that

$$ \sigma = \sigma_0 + \partial\Delta. $$

We have

$$ \mathcal{A}_n(\langle\sigma, \phi\rangle) = \int_{\sigma_0} 3c \mathcal{L}^{\uparrow}\phi + \int_{\partial\Delta} 3c \mathcal{L}^{\uparrow}\phi + O(\epsilon_n \|\sigma\|_F). \quad (70) $$

Consider an edge $e \in \sigma_0$. Since

$$ \int_e 3c \, de = 3 \left( \int_0^{\epsilon_n} \frac{x^2}{\epsilon_n^2} dx + \int_{\epsilon_n}^{2\epsilon_n} \frac{(2\epsilon_n - x)^2}{\epsilon_n^2} dx \right) = 2\epsilon_n = \int_e de, $$

we can use a Taylor expansion on $\mathcal{L}^{\uparrow}\phi$ at the barycenter of $e$ which we denote by $G$ to obtain

$$ \int_e 3c \mathcal{L}^{\uparrow}\phi = \int_e 3c \mathcal{L}^{\uparrow}\phi(G) + O(\epsilon_n^3) = \int_e \mathcal{L}^{\uparrow}\phi(G) + O(\epsilon_n^3) = \int_e \mathcal{L}^{\uparrow}\phi + O(\epsilon_n^3). $$

This yields for the first term in the right hand side of (70) that

$$ \int_{\sigma_0} 3c \mathcal{L}^{\uparrow}\phi = \int_{\sigma_0} \mathcal{L}^{\uparrow}\phi + O(\epsilon_n^2 \|\sigma_0\|_{\mathcal{C}^1}). $$

Now let us consider the second term in the right hand side of (70). For a smooth kernel $K$ on $\mathbb{R}^2$, define:

$$ K_n(x) = \frac{1}{h_n^2} K\left(\frac{x}{h_n}\right). $$

Then, we have

$$ \int_{\partial\Delta} 3c \mathcal{L}^{\uparrow}\phi = \int_{\partial\Delta} 3(K_n * c) \mathcal{L}^{\uparrow}\phi + \int_{\partial\Delta} 3(c - K_n * c) \mathcal{L}^{\uparrow}\phi. \quad (71) $$

First, remark that for $h_n$ sufficiently small, $\|c - K_n * c\|_{\infty} \le \epsilon_n^2$. So for the second term in the rhs of (71),

$$ \int_{\partial\Delta} 3(c - K_n * c) \mathcal{L}^{\uparrow}\phi = O(\epsilon_n^2 \|\Delta\|_{\mathcal{C}^2}). $$

Let us consider the first term in the rhs of (71). Since $(K_n * c)\mathcal{L}^{\uparrow}\phi$ is a $\mathcal{C}^1$ differential form (thanks to the smoothing of $c$ by $K_n$), we can use Stokes' Theorem to obtain

$$ \begin{aligned} \int_{\partial\Delta} 3(K_n * c) \mathcal{L}^{\uparrow}\phi &= 3 \int_{\Delta} d((K_n * c) \mathcal{L}^{\uparrow}\phi) \\ &= 3 \int_{\Delta} (K_n * c) d\mathcal{L}^{\uparrow}\phi + 3 \int_{\Delta} \left( \frac{\partial K_n * c}{\partial x_1} (\mathcal{L}^{\uparrow}\phi)^2 - \frac{\partial K_n * c}{\partial x_2} (\mathcal{L}^{\uparrow}\phi)^1 \right) dx_1 dx_2, \end{aligned} \quad (72) $$

by using (46).

The computation now depends on details shown in Lemma 47 in Appendix (and leading to (73) and (74)). Let $\tau$ be a triangle of $\Delta$ with barycenter denoted again by $G$. By a Taylor expansion of $d\mathcal{L}^{\uparrow}\phi$ at $G$, we have

$$ \begin{aligned} \int_{\tau} 3(K_n * c) d\mathcal{L}^{\uparrow}\phi &= \int_{\tau} 3(K_n * c) d\mathcal{L}^{\uparrow}\phi(G) + O(\epsilon_n^4) \\ &= \frac{5}{3} \int_{\tau} d\mathcal{L}^{\uparrow}\phi(G) + O(\epsilon_n^4). \end{aligned} \quad (73) $$
Now, let $\sigma_0$ and $\Delta$ be a cycle and a 2-chain (in the simplicial complex $C_n$) such that

$$ \sigma = \sigma_0 + \partial\Delta. $$

We have

$$ A_n(\langle\sigma, \phi\rangle) = \int_{\sigma_0} 3c \mathcal{L}^{\uparrow}\phi + \int_{\partial\Delta} 3c \mathcal{L}^{\uparrow}\phi + O(\epsilon_n \|\sigma\|_F). \quad (70) $$

Consider an edge $e \in \sigma_0$. Since

$$ \int_e 3c \, de = 3 \left( \int_0^{\epsilon_n} \frac{x^2}{\epsilon_n^2} dx + \int_{\epsilon_n}^{2\epsilon_n} \frac{(2\epsilon_n - x)^2}{\epsilon_n^2} dx \right) = 2\epsilon_n = \int_e de, $$

we can use a Taylor expansion on $\mathcal{L}^{\uparrow}\phi$ at the barycenter of $e$ which we denote by $G$ to obtain

$$ \int_e 3c \mathcal{L}^{\uparrow}\phi = \int_e 3c \mathcal{L}^{\uparrow}\phi(G) + O(\epsilon_n^3) = \int_e \mathcal{L}^{\uparrow}\phi(G) + O(\epsilon_n^3) = \int_e \mathcal{L}^{\uparrow}\phi + O(\epsilon_n^3). $$

This yields for the first term in the right hand side of (70) that

$$ \int_{\sigma_0} 3c \mathcal{L}^{\uparrow}\phi = \int_{\sigma_0} \mathcal{L}^{\uparrow}\phi + O(\epsilon_n^2 \|\sigma_0\|_{\mathcal{C}^1}). $$

Now let us consider the second term in the right hand side of (70). For a smooth kernel $K$ on $\mathbb{R}^2$, define:

$$ K_n(x) = \frac{1}{h_n^2} K\left(\frac{x}{h_n}\right). $$

Then, we have

$$ \int_{\partial\Delta} 3c \mathcal{L}^{\uparrow}\phi = \int_{\partial\Delta} 3(K_n * c)\mathcal{L}^{\uparrow}\phi + \int_{\partial\Delta} 3(c - K_n * c)\mathcal{L}^{\uparrow}\phi. \quad (71) $$

First, remark that for $h_n$ sufficiently small, $\|c - K_n * c\|_{\infty} \le \epsilon_n^2$. So for the second term in the rhs of (71),

$$ \int_{\partial\Delta} 3(c - K_n * c) \mathcal{L}^{\uparrow}\phi = O(\epsilon_n^2 \|\Delta\|_{\mathcal{C}^2}). $$

Let us consider the first term in the rhs of (71). Since $(K_n * c)\mathcal{L}^{\uparrow}\phi$ is a $\mathcal{C}^1$ differential form (thanks to the smoothing of $c$ by $K_n$), we can use Stokes' Theorem to obtain

$$ \begin{aligned} \int_{\partial\Delta} 3(K_n * c) \mathcal{L}^{\uparrow}\phi &= 3 \int_{\Delta} d((K_n * c) \mathcal{L}^{\uparrow}\phi) \\ &= 3 \int_{\Delta} (K_n * c) d\mathcal{L}^{\uparrow}\phi + 3 \int_{\Delta} \left( \frac{\partial K_n * c}{\partial x_1} (\mathcal{L}^{\uparrow}\phi)^2 - \frac{\partial K_n * c}{\partial x_2} (\mathcal{L}^{\uparrow}\phi)^1 \right) dx_1 dx_2, \end{aligned} \quad (72) $$

by using (46).

The computation now depends on details shown in Lemma 47 in Appendix (and leading to (73) and (74)). Let $\tau$ be a triangle of $\Delta$ with barycenter denoted again by $G$. By a Taylor expansion of $d\mathcal{L}^{\uparrow}\phi$ at $G$, we have

$$ \begin{aligned} \int_{\tau} 3(K_n * c) d\mathcal{L}^{\uparrow}\phi &= \int_{\tau} 3(K_n * c) d\mathcal{L}^{\uparrow}\phi(G) + O(\epsilon_n^4) \\ &= \frac{5}{3} \int_{\tau} d\mathcal{L}^{\uparrow}\phi(G) + O(\epsilon_n^4). \end{aligned} \quad (73) $$
Furthermore, taking $h_n$ sufficiently small and using another Taylor expansion for $(\mathcal{L}^\uparrow\phi)^1$ and $(\mathcal{L}^\uparrow\phi)^2$ at $G = (G_1, G_2)$, we obtain, using Lemma 47,

$$ 
\begin{aligned}
& 3 \int_{\tau} \left( \frac{\partial K_n * c}{\partial x_1} (\mathcal{L}^{\uparrow}\phi)^2 - \frac{\partial K_n * c}{\partial x_2} (\mathcal{L}^{\uparrow}\phi)^1 \right) dx_1 dx_2 \\
&= 3 \frac{\partial}{\partial x_1} (\mathcal{L}^{\uparrow}\phi)^2 (G) \int_{\tau} (x_1 - G_1) \frac{\partial K_n * c}{\partial x_1} dx_1 dx_2 - 3 \frac{\partial}{\partial x_2} (\mathcal{L}^{\uparrow}\phi)^1 (G) \int_{\tau} (x_2 - G_2) \frac{\partial K_n * c}{\partial x_2} dx_1 dx_2 + O(\epsilon_n^4) \\
&= -\frac{2}{3} \int_{\tau} d\mathcal{L}^{\uparrow}\phi(G) + O(\epsilon_n^4).
\end{aligned} 
\quad (74) $$

From (72), (73) and (74), and using another Taylor expansion around $G$,

$$ 3 \int_{\tau} d((K_n * c)\mathcal{L}^{\uparrow}\phi) = \int_{\tau} d\mathcal{L}^{\uparrow}\phi(G) + O(\epsilon_n^4) = \int_{\tau} d\mathcal{L}^{\uparrow}\phi + O(\epsilon_n^4). $$

Hence, using Stokes's theorem once more,

$$ \int_{\partial\Delta} 3c\mathcal{L}^{\uparrow}\phi = \int_{\partial\Delta} \mathcal{L}^{\uparrow}\phi + O(\epsilon_n^2 \|\Delta\|_{\varepsilon^2}). $$

Finally, the result follows from taking the $\sigma_0$ and $\Delta$ achieving the flat norm. ■

### 4.5.3 Tightness of the sequence ($X^{(n)}$)

The main result of this section is:

**Proposition 40.** Let us work under the assumptions of Theorem 36. The distributions of the $X^{(n)}$'s form a tight family in the set of probability measures on $D(\mathbb{R}_+, \mathcal{C}_{1,4})$.

Once Proposition 40 is proved, since $\mathcal{C}_{1,4}$ is Polish, we will obtain that the sequence of distributions of $X^n$'s is relatively compact for the weak convergence, by Prohorov's theorem [7, Th.5.1 P.59].

*Proof of Proposition 40.* First recall that as a direct consequence of the Banach-Alaoglu theorem, see e.g. [45] or [9, Th. III.15], the unit ball in $\mathcal{C}_{1,4}$ is a compact set with respect to the weak-* topology. This and Proposition 35 hence imply that the marginals of $X^{(n)}$ are tight. Thus, it only remains to check the tightness of the distributions of the real-valued càdlàg processes $F(X^{(n)})$ for test functions $F$ of the form (44) (see [28, Theorem 3.1]). For this, we can use Aldous-Rebolledo's criterion [2, 29].

Starting from (50), we have:

$$ 
\begin{aligned}
F(\langle X_{\epsilon_n^{-2}t}^n, \phi \rangle) &= F(\langle X_{\epsilon_n^{-2}s}^n, \phi \rangle) \\
&\quad + \int_{\epsilon_n^{-2}s}^{\epsilon_n^{-2}t} \int_{\mathbb{R}_+} \int_{S_n^2} \left( F(\langle X_{u_-}^n, \phi \rangle - \langle \partial_2\tau, \phi \rangle) - F(\langle X_{u_-}^n, \phi \rangle) \right) \mathbf{1}_{\{\theta \le w(X_{u_-}^n, \partial_2\tau)\}} N(du, d\theta, d\tau) \\
&= F(\langle X_s^{(n)}, \phi \rangle) + A_{\epsilon_n^{-2}t}^n - A_{\epsilon_n^{-2}s}^n + M_{\epsilon_n^{-2}t}^n - M_{\epsilon_n^{-2}s}^n,
\end{aligned} 
\quad (75) $$

where $A^n$ is a predictable finite variation process

$$ A_{\epsilon_n^{-2}t}^{(n)} = \epsilon_n^{-2} \int_0^t \sum_{\tau \in S_n^2} \left( F(\langle X_u^{(n)}, \phi \rangle - \langle \partial_2 \tau, \phi \rangle) - F(\langle X_u^{(n)}, \phi \rangle) \right) w(X_u^{(n)}, \partial_2 \tau) du, \quad (76) $$

and where $M^n$ is a square integrable martingale started from 0 and with quadratic variation:

$$ \langle M^n \rangle_{\epsilon_n^{-2}t} = \epsilon_n^{-2} \int_0^t \sum_{\tau \in S_n^2} \left( F(\langle X_u^{(n)}, \phi \rangle - \langle \partial_2 \tau, \phi \rangle) - F(\langle X_u^{(n)}, \phi \rangle) \right)^2 w(X_u^{(n)}, \partial_2 \tau) du. \quad (77) $$
Furthermore, taking $h_n$ sufficiently small and using another Taylor expansion for $(\mathcal{L}^{\uparrow}\phi)^1$ and $(\mathcal{L}^{\uparrow}\phi)^2$ at $G = (G_1, G_2)$, we obtain, using Lemma 47,

$$ 
\begin{aligned}
& 3 \int_{\tau} \left( \frac{\partial K_n * c}{\partial x_1} (\mathcal{L}^{\uparrow}\phi)^2 - \frac{\partial K_n * c}{\partial x_2} (\mathcal{L}^{\uparrow}\phi)^1 \right) dx_1 dx_2 \\
&= 3 \frac{\partial}{\partial x_1} (\mathcal{L}^{\uparrow}\phi)^2 (G) \int_{\tau} (x_1 - G_1) \frac{\partial K_n * c}{\partial x_1} dx_1 dx_2 - 3 \frac{\partial}{\partial x_2} (\mathcal{L}^{\uparrow}\phi)^1 (G) \int_{\tau} (x_2 - G_2) \frac{\partial K_n * c}{\partial x_2} dx_1 dx_2 + O(\epsilon_n^4) \\
&= -\frac{2}{3} \int_{\tau} d\mathcal{L}^{\uparrow}\phi(G) + O(\epsilon_n^4).
\end{aligned} 
\quad (74) $$

From (72), (73) and (74), and using another Taylor expansion around $G$,

$$ 3 \int_{\tau} d((K_n * c)\mathcal{L}^{\uparrow}\phi) = \int_{\tau} d\mathcal{L}^{\uparrow}\phi(G) + O(\epsilon_n^4) = \int_{\tau} d\mathcal{L}^{\uparrow}\phi + O(\epsilon_n^4). $$

Hence, using Stokes's theorem once more,

$$ \int_{\partial\Delta} 3c\mathcal{L}^{\uparrow}\phi = \int_{\partial\Delta} \mathcal{L}^{\uparrow}\phi + O(\epsilon_n^2 \|\Delta\|_{\varepsilon^2}). $$

Finally, the result follows from taking the $\sigma_0$ and $\Delta$ achieving the flat norm. ■

### 4.5.3 Tightness of the sequence ($X^{(n)}$)

The main result of this section is:

**Proposition 40.** Let us work under the assumptions of Theorem 36. The distributions of the $X^{(n)}$'s form a tight family in the set of probability measures on $D(\mathbb{R}_+, \mathcal{C}_{1,4})$.

Once Proposition 40 is proved, since $\mathcal{C}_{1,4}$ is Polish, we will obtain that the sequence of distributions of $X^n$'s is relatively compact for the weak convergence, by Prohorov's theorem [7, Th.5.1 P.59].

*Proof of Proposition 40.* First recall that as a direct consequence of the Banach-Alaoglu theorem, see e.g. [45] or [9, Th. III.15], the unit ball in $\mathcal{C}_{1,4}$ is a compact set with respect to the weak-* topology. This and Proposition 35 hence imply that the marginals of $X^{(n)}$ are tight. Thus, it only remains to check the tightness of the distributions of the real-valued càdlàg processes $F(X^{(n)})$ for test functions $F$ of the form (44) (see [28, Theorem 3.1]). For this, we can use Aldous-Rebolledo's criterion [2, 29].

Starting from (50), we have:

$$ 
\begin{aligned}
F(\langle X_{\epsilon_n^{-2}t}^n, \phi \rangle) &= F(\langle X_{\epsilon_n^{-2}s}^n, \phi \rangle) \\
&\quad + \int_{\epsilon_n^{-2}s}^{\epsilon_n^{-2}t} \int_{\mathbb{R}_+} \int_{S_n^2} \left( F(\langle X_{u_-}^n, \phi \rangle - \langle \partial_2 \tau, \phi \rangle) - F(\langle X_{u_-}^n, \phi \rangle) \right) \mathbf{1}_{\{\theta \le w(X_{u_-}^n, \partial_2 \tau)\}} N(du, d\theta, d\tau) \\
&= F(\langle X_s^{(n)}, \phi \rangle) + A_{\epsilon_n^{-2}t}^n - A_{\epsilon_n^{-2}s}^n + M_{\epsilon_n^{-2}t}^n - M_{\epsilon_n^{-2}s}^n,
\end{aligned} 
\quad (75) $$

where $A^n$ is a predictable finite variation process

$$ A_{\epsilon_n^{-2}t}^{(n)} = \epsilon_n^{-2} \int_0^t \sum_{\tau \in S_n^2} \left( F(\langle X_u^{(n)}, \phi \rangle - \langle \partial_2 \tau, \phi \rangle) - F(\langle X_u^{(n)}, \phi \rangle) \right) w(X_u^{(n)}, \partial_2 \tau) du, \quad (76) $$

and where $M^n$ is a square integrable martingale started from 0 and with quadratic variation:

$$ \langle M^n \rangle_{\epsilon_n^{-2}t} = \epsilon_n^{-2} \int_0^t \sum_{\tau \in S_n^2} \left( F(\langle X_u^{(n)}, \phi \rangle - \langle \partial_2 \tau, \phi \rangle) - F(\langle X_u^{(n)}, \phi \rangle) \right)^2 w(X_u^{(n)}, \partial_2 \tau) du. \quad (77) $$