Aldous-Rebolledo's criterion tells us to show that the real-valued processes $(A_{\epsilon_n^{-2}t}^n)$ and $((M^n)_{\epsilon_n^{-2}t})$ satisfy the Aldous criterion [2]. Let $\delta > 0$ and let $\sigma$ and $\sigma'$ be two stopping times such that $\sigma \le \sigma' \le \sigma + \delta$. For $\eta > 0$, by Markov's inequality and Proposition 39:

$$ 
\begin{aligned}
\mathbf{P}(|A_{\epsilon_n^{-2}\sigma'}^n - A_{\epsilon_n^{-2}\sigma}^n| \ge \eta) &\le \frac{1}{\eta} \left( \mathbb{E}\left[ \int_{\sigma}^{\sigma'} F'((X_u^{(n)}, \phi)) A_n((\cdot, \phi))(X_u^{(n)}) \, du \right] + \mathbb{E}[|R_n(\sigma, \sigma')|] \right) \\
&\le \frac{1}{\eta} \left( \mathbb{E}\left[ \int_{\sigma}^{\sigma'} \|F'\|_{\infty} \left( |\langle \mathcal{L}^{\uparrow} \phi, X_u^{(n)} \rangle| + C_1 \epsilon_n \|X_u^{(n)}\|_F \right) \, du \right] + \mathbb{E}[|R_n(\sigma, \sigma')|] \right)
\end{aligned}
\quad (78) $$

where for $s$ and $t$

$$ 
\begin{aligned}
|R_n(s, t)| &= \left| \frac{1}{2} \epsilon_n^{-2} \int_s^t \sum_{\tau \in S_2^n} \langle \partial_{2\tau}, \phi \rangle^2 F''(\langle X_u^{(n)}, \phi \rangle - \theta_\tau \langle \partial_{2\tau}, \phi \rangle) w(X_u^{(n)}, \partial_{2\tau}) du \right| \\
&\le C_2 \|F''\|_{\infty} \sup_{s \le u \le t} (\epsilon_n \|X_u^{(n)}\|_{\mathcal{C}_1}) \times |t-s| \\
&\le C_2 \|F''\|_{\infty} \sup_{s \le u \le t} (\|X_u^{(n)}\|_F) \times |t-s|.
\end{aligned}
$$

Now, using that for $\phi \in \mathcal{C}^{1,4}$, $\mathcal{L}^{\uparrow}\phi \in \mathcal{C}^{1,2}$ and that $\mathcal{C}_{1,2} \subset \mathcal{C}_{1,4}$,

$$ |\langle \mathcal{L}^{\uparrow} \phi, X_u^{(n)} \rangle| \le \|X_u^{(n)}\|_{\mathcal{C}_{1,2}} \|\mathcal{L}^{\uparrow} \phi\|_{\mathcal{C}^{1,2}} \le C \|X_u^{(n)}\|_{\mathcal{C}_{1,4}} \|\phi\|_{\mathcal{C}^{1,4}}, $$

for a constant $C$ (that will change from line to line). Under the Conjecture 33, and by Proposition 35,

$$ \mathbf{P}(|\langle A_{\sigma'}^n, \phi \rangle - \langle A_{\sigma}^n, \phi \rangle| \ge \eta) \le \frac{C\delta}{\eta} \quad (79) $$

that can be made as small as wished by setting $\delta$ sufficiently small.

For the predictable quadratic variation process of $\langle M^n \rangle$, using similar computation with a Taylor expansion at the order 1, we have:

$$ 
\begin{aligned}
\mathbb{E}(|\langle M^n \rangle_{\epsilon_n^{-2}t} - \langle M^n \rangle_{\epsilon_n^{-2}s}|) &\le C \epsilon_n^{-2} \int_s^t \mathbb{E} \left[ \sum_{\tau \in S_2^n} (\epsilon_n^4 \|F'\|_{\infty}^2) w(X_u^{(n)}, \partial_{2\tau}) \, du \right] \\
&\le C \delta \|F'\|_{\infty}^2 \sup_{s \le u \le t} \|X_u^{(n)}\|_F.
\end{aligned}
\quad (80) $$

Under Conjecture 33,

$$ \mathbf{P}(|\langle M^n \rangle_{\epsilon_n^{-2}\sigma'} - \langle M^n \rangle_{\epsilon_n^{-2}\sigma}| \ge \eta) \le \frac{C}{\eta} \|F'\|_{\infty}^2 \times \delta. \quad (81) $$

This concludes the proof of Proposition 40 and of Theorem 36. ■

### 4.5.4 Convergence of the generators (2)

Let us consider a limiting value $\bar{X}$ of the tight sequence $(X^{(n)})$ in $\mathbb{D}(\mathbf{R}_+, \mathcal{C}_{1,4})$ and a subsequence (again denoted by $(X^{(n)})$ for the sake of notation) that converges in distribution to $\bar{X}$. By Skorokhod representation theorem (see e.g. [6, Th. 6.7 p.70]) it is possible to assume that the convergence is almost sure.

Notice that the sequence $(X^{(n)})$ is in fact C-tight (see [27, p.561]) as for all $T > 0$ and for all $\eta > 0$,

$$ \lim_{n \to +\infty} \mathbf{P} \left( \sup_{t \le T} \|X_t^{(n)} - X_{t-}^{(n)}\|_{\mathcal{C}_{1,4}} > \eta \right) = \lim_{n \to +\infty} \mathbf{P} \left( \|\partial_2[012]\|_{\mathcal{C}_{1,4}} > \eta \right) = 0, \quad (82) $$
Aldous-Rebolledo's criterion tells us to show that the real-valued processes $(A_{\epsilon_n^{-2}t}^n)$ and $((M^n)_{\epsilon_n^{-2}t})$ satisfy the Aldous criterion [2]. Let $\delta > 0$ and let $\sigma$ and $\sigma'$ be two stopping times such that $\sigma \le \sigma' \le \sigma + \delta$. For $\eta > 0$, by Markov's inequality and Proposition 39:

$$ 
\begin{aligned}
\mathbf{P}(|A_{\epsilon_n^{-2}\sigma'}^n - A_{\epsilon_n^{-2}\sigma}^n| \ge \eta) &\le \frac{1}{\eta} \left( \mathbb{E}\left[ \int_{\sigma}^{\sigma'} F'((X_u^{(n)}, \phi)) A_n((\cdot, \phi))(X_u^{(n)}) \, du \right] + \mathbb{E}[|R_n(\sigma, \sigma')|] \right) \\
&\le \frac{1}{\eta} \left( \mathbb{E}\left[ \int_{\sigma}^{\sigma'} \|F'\|_{\infty} \left( |\langle \mathcal{L}^{\uparrow} \phi, X_u^{(n)} \rangle| + C_1 \epsilon_n \|X_u^{(n)}\|_F \right) \, du \right] + \mathbb{E}[|R_n(\sigma, \sigma')|] \right)
\end{aligned}
\quad (78) $$

where for $s$ and $t$

$$ 
\begin{aligned}
|R_n(s,t)| &= \left| \frac{1}{2} \epsilon_n^{-2} \int_s^t \sum_{\tau \in S_2^n} \langle \partial_{2\tau}, \phi \rangle^2 F''(\langle X_u^{(n)}, \phi \rangle - \theta_{\tau} \langle \partial_{2\tau}, \phi \rangle) w(X_u^{(n)}, \partial_{2\tau}) du \right| \\
&\le C_2 \|F''\|_{\infty} \sup_{s \le u \le t} (\epsilon_n \|X_u^{(n)}\|_{\mathcal{C}_1}) \times |t-s| \\
&\le C_2 \|F''\|_{\infty} \sup_{s \le u \le t} (\|X_u^{(n)}\|_F) \times |t-s|.
\end{aligned}
$$

Now, using that for $\phi \in \mathcal{C}^{1,4}$, $\mathcal{L}^{\uparrow}\phi \in \mathcal{C}^{1,2}$ and that $\mathcal{C}_{1,2} \subset \mathcal{C}_{1,4}$,

$$ |\langle \mathcal{L}^{\uparrow} \phi, X_u^{(n)} \rangle| \le \|X_u^{(n)}\|_{\mathcal{C}_{1,2}} \|\mathcal{L}^{\uparrow} \phi\|_{\mathcal{C}^{1,2}} \le C \|X_u^{(n)}\|_{\mathcal{C}_{1,4}} \|\phi\|_{\mathcal{C}^{1,4}}, $$

for a constant $C$ (that will change from line to line). Under the Conjecture 33, and by Proposition 35,

$$ \mathbf{P}(|\langle A_{\sigma'}^n, \phi \rangle - \langle A_{\sigma}^n, \phi \rangle| \ge \eta) \le \frac{C\delta}{\eta} \quad (79) $$

that can be made as small as wished by setting $\delta$ sufficiently small.

For the predictable quadratic variation process of $\langle M^n \rangle$, using similar computation with a Taylor expansion at the order 1, we have:

$$ 
\begin{aligned}
\mathbb{E}(|\langle M^n \rangle_{\epsilon_n^{-2}t} - \langle M^n \rangle_{\epsilon_n^{-2}s}|) &\le C \epsilon_n^{-2} \int_s^t \mathbb{E} \left[ \sum_{\tau \in S_2^n} (\epsilon_n^4 \|F'\|_{\infty}^2) w(X_u^{(n)}, \partial_{2\tau}) \, du \right] \\
&\le C \delta \|F'\|_{\infty}^2 \sup_{s \le u \le t} \|X_u^{(n)}\|_F.
\end{aligned}
\quad (80) $$

Under Conjecture 33,

$$ \mathbf{P}(|\langle M^n \rangle_{\epsilon_n^{-2}\sigma'} - \langle M^n \rangle_{\epsilon_n^{-2}\sigma}| \ge \eta) \le \frac{C}{\eta} \|F'\|_{\infty}^2 \times \delta. \quad (81) $$

This concludes the proof of Proposition 40 and of Theorem 36. ■

### 4.5.4 Convergence of the generators (2)

Let us consider a limiting value $\bar{X}$ of the tight sequence $(X^{(n)})$ in $\mathbb{D}(\mathbf{R}_+, \mathcal{C}_{1,4})$ and a subsequence (again denoted by $(X^{(n)})$ for the sake of notation) that converges in distribution to $\bar{X}$. By Skorokhod representation theorem (see e.g. [6, Th. 6.7 p.70]) it is possible to assume that the convergence is almost sure.

Notice that the sequence $(X^{(n)})$ is in fact C-tight (see [27, p.561]) as for all $T > 0$ and for all $\eta > 0$,

$$ \lim_{n \to +\infty} \mathbf{P} \left( \sup_{t \le T} \| X_t^{(n)} - X_{t-}^{(n)} \|_{\mathcal{C}_{1,4}} > \eta \right) = \lim_{n \to +\infty} \mathbf{P} \left( \| \partial_2 [012] \|_{\mathcal{C}_{1,4}} > \eta \right) = 0, \quad (82) $$
where [012] stands for the triangle in Fig. 6 whose norm is of order $\epsilon_n$ and smaller than $\eta$ for $n$ sufficiently large. The limiting values are then in $C(\mathbf{R}_+, \mathfrak{C}_{1,4})$.

Recall that the unit ball of $\mathfrak{C}_{1,4}$ is compact, and hence we can find a denumerable dense family of functions $(\phi_k)_{k \in \mathbb{N}}$ of $\mathfrak{C}^{1,4}$.

**Proposition 41.** Let us work under the assumptions of Theorem 36. For $F(x) = x$ and for $\phi_k \in \mathfrak{C}^{1,4}$, $k \in \mathbb{N}$, as introduced above the Proposition, let us denote by $M^{n,k}$ the martingale defined in (75). This real-valued martingale is tight and converges in distribution to the martingale $W^{(k)} \in C(\mathbf{R}_+, \mathbf{R})$ started at zero and with predictable quadratic variation:

$$ \langle W^{(k)} \rangle_t := \lim_{n \to +\infty} \epsilon_n^{-2} \int_0^t \sum_{\tau \in S_2^n} \langle \partial_2 \tau, \phi_k \rangle^2 w(X_u^{(n)}, \partial_2 \tau) du. \quad (83) $$

Moreover, it is possible to define a martingale measure $(W_t(d\sigma))_{t \in \mathbf{R}_+}$ such that for all $k \in \mathbb{N}$,

$$ W_t^{(k)} = \int_{\mathfrak{C}_{1,4}} \phi_k(\sigma) W_t(d\sigma). $$

*Proof.* Adapting the computation of the previous section, we can show that not only the sequence $(X^{(n)})$ is tight, but also the joint sequence $(X^{(n)}, M^{n,k}, k \in \mathbb{N})$. Let us consider a limiting value $(\bar{X}, W^{(k)}, k \in \mathbb{N})$ of the tight sequence $(X^{(n)}, M^{n,k}, k \in \mathbb{N})$ and a subsequence (again denoted by $(X^{(n)}, M^{n,k}, k \in \mathbb{N})$ for the sake of notation) that converges in distribution to $(\bar{X}, W^{(k)}, k \in \mathbb{N})$. We conclude by applying [27, Theorem 2.4 P.487]. By the Skorokhod representation theorem (see e.g. [6, Th. 6.7 p.70]) it is possible to assume that the convergence is almost sure. Linearity then implies the existence of a martingale measure (see [55]). ■

Recall now that the test functions characterizing the convergence of our generators have been defined in (44). For $\phi \in \mathfrak{C}^1$ and $F \in C(\mathbf{R})$, we will denote $F_\phi(\sigma) = F(\langle \sigma, \phi \rangle)$.

**Corollary 42.** Let us work under the assumptions of Theorem 36. Consider the martingale $M^n$ defined for any functions $F \in C_b^2(\mathbf{R}, \mathbf{R})$ and $\phi$ in (75). The sequence of real-valued martingales $(M^n)$ converges in distribution to the centered martingale $W \in C(\mathbf{R}_+, \mathbf{R})$ with predictable quadratic variation:

$$ \langle W \rangle_t := \int_0^t F'^2(\langle \bar{X}_u, \phi \rangle) d\langle W_u, \phi \rangle. \quad (84) $$

*Proof.* Recall the bracket of the martingale $M^n$ computed in (77). We have:

$$ \begin{aligned} \lim_{n \to +\infty} \langle M^n \rangle_{\epsilon_n^{-2} t} &= \lim_{n \to +\infty} \epsilon_n^{-2} \int_0^t \sum_{\tau \in S_2^n} \left( F(\langle X_u^{(n)}, \phi \rangle - \langle \partial_2 \tau, \phi \rangle) - F(\langle X_u^{(n)}, \phi \rangle) \right)^2 w(X_u^{(n)}, \partial_2 \tau) du \\ &= \lim_{n \to +\infty} \epsilon_n^{-2} \int_0^t F'^2(\langle X_u^{(n)}, \phi \rangle) \sum_{\tau \in S_2^n} \langle \partial_2 \tau, \phi \rangle^2 w(X_u^{(n)}, \partial_2 \tau) du + O(\epsilon_n^4 \sup_{u \le t} \|X_u^{(n)}\|_F), \end{aligned} $$

by using Taylor's formula as in (78). The result is then a consequence of Conjecture 33 and of (83). ■

We are now almost in position to show that the limiting solution $X$ solves the martingale problem associated with $\mathfrak{A}$. Notice that in (61), we have several equivalent formulation of the last term. This is stated in the following corollary, whose proof is postponed to Appendix C. The proof is similar to the one of Proposition 39 and based on Taylor expansions.
where [012] stands for the triangle in Fig. 6 whose norm is of order $\epsilon_n$ and smaller than $\eta$ for $n$ sufficiently large. The limiting values are then in $C(\mathbf{R}_+, \mathfrak{C}_{1,4})$.

Recall that the unit ball of $\mathfrak{C}_{1,4}$ is compact, and hence we can find a denumerable dense family of functions $(\phi_k)_{k \in \mathbb{N}}$ of $\mathfrak{C}^{1,4}$.

**Proposition 41.** Let us work under the assumptions of Theorem 36. For $F(x) = x$ and for $\phi_k \in \mathfrak{C}^{1,4}$, $k \in \mathbb{N}$, as introduced above the Proposition, let us denote by $M^{n,k}$ the martingale defined in (75). This real-valued martingale is tight and converges in distribution to the martingale $W^{(k)} \in C(\mathbf{R}_+, \mathbf{R})$ started at zero and with predictable quadratic variation:

$$ \langle W^{(k)} \rangle_t := \lim_{n \to +\infty} \epsilon_n^{-2} \int_0^t \sum_{\tau \in S_2^n} \langle \partial_2 \tau, \phi_k \rangle^2 w(X_u^{(n)}, \partial_2 \tau) du. \quad (83) $$

Moreover, it is possible to define a martingale measure $(W_t(d\sigma))_{t \in \mathbf{R}_+}$ such that for all $k \in \mathbb{N}$,

$$ W_t^{(k)} = \int_{\mathfrak{C}_{1,4}} \phi_k(\sigma) W_t(d\sigma). $$

*Proof.* Adapting the computation of the previous section, we can show that not only the sequence $(X^{(n)})$ is tight, but also the joint sequence $(X^{(n)}, M^{n,k}, k \in \mathbb{N})$. Let us consider a limiting value $(\bar{X}, W^{(k)}, k \in \mathbb{N})$ of the tight sequence $(X^{(n)}, M^{n,k}, k \in \mathbb{N})$ and a subsequence (again denoted by $(X^{(n)}, M^{n,k}, k \in \mathbb{N})$ for the sake of notation) that converges in distribution to $(\bar{X}, W^{(k)}, k \in \mathbb{N})$. We conclude by applying [27, Theorem 2.4 P.487]. By the Skorokhod representation theorem (see e.g. [6, Th. 6.7 p.70]) it is possible to assume that the convergence is almost sure. Linearity then implies the existence of a martingale measure (see [55]). ■

Recall now that the test functions characterizing the convergence of our generators have been defined in (44). For $\phi \in \mathfrak{C}^1$ and $F \in C(\mathbf{R})$, we will denote $F_\phi(\sigma) = F(\langle \sigma, \phi \rangle)$.

**Corollary 42.** Let us work under the assumptions of Theorem 36. Consider the martingale $M^n$ defined for any functions $F \in C_b^2(\mathbf{R}, \mathbf{R})$ and $\phi$ in (75). The sequence of real-valued martingales $(M^n)$ converges in distribution to the centered martingale $W \in C(\mathbf{R}_+, \mathbf{R})$ with predictable quadratic variation:

$$ \langle W \rangle_t := \int_0^t F'^2(\langle \bar{X}_u, \phi \rangle) d\langle W_u, \phi \rangle. \quad (84) $$

*Proof.* Recall the bracket of the martingale $M^n$ computed in (77). We have:

$$ \begin{aligned} \lim_{n \to +\infty} \langle M^n \rangle_{\epsilon_n^{-2} t} &= \lim_{n \to +\infty} \epsilon_n^{-2} \int_0^t \sum_{\tau \in S_2^n} \left( F(\langle X_u^{(n)}, \phi \rangle - \langle \partial_2 \tau, \phi \rangle) - F(\langle X_u^{(n)}, \phi \rangle) \right)^2 w(X_u^{(n)}, \partial_2 \tau) du \\ &= \lim_{n \to +\infty} \epsilon_n^{-2} \int_0^t F'^2(\langle X_u^{(n)}, \phi \rangle) \sum_{\tau \in S_2^n} \langle \partial_2 \tau, \phi \rangle^2 w(X_u^{(n)}, \partial_2 \tau) du + O(\epsilon_n^4 \sup_{u \le t} \|X_u^{(n)}\|_F), \end{aligned} $$

by using Taylor's formula as in (78). The result is then a consequence of Conjecture 33 and of (83). ■

We are now almost in position to show that the limiting solution $X$ solves the martingale problem associated with $\mathfrak{A}$. Notice that in (61), we have several equivalent formulation of the last term. This is stated in the following corollary, whose proof is postponed to Appendix C. The proof is similar to the one of Proposition 39 and based on Taylor expansions.
**Corollary 43.** For $X_u \in \mathfrak{C}_{1,4}$, recall the definition of $\tilde{X}_u$ in (59). We have:

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

**Proposition 44.** Let us consider $\phi \in \mathfrak{C}_{1,4}$ and $F \in C_b^3(\mathbb{R})$ and recall the definition of $\mathfrak{A}$ in (61). Then, for $k \in \mathbb{N}$, for $0 \le t_1 \le \dots t_k \le s < t$, and $\Phi_1, \dots, \Phi_k$ bounded continuous functions of $\mathfrak{C}_{1,4}$,

$$ \lim_{n \to +\infty} \mathbb{E} \left[ \left( F(\langle X_t^{(n)}, \phi \rangle) - F(\langle X_s^{(n)}, \phi \rangle) - \int_s^t \mathfrak{A} F_\phi(X_u^{(n)}) du \right) \prod_{i=1}^k \Phi_i(X_{t_i}^{(n)}) \right] = 0. \quad (85) $$

*Proof.* Let us consider $s < t$. Using (50) and It's formula [26, Th.5.1 P.66] for jump processes, we have that for every $n \in \mathbb{N}$:

$$ \mathbb{E} \left[ \left( F(\langle X_t^{(n)}, \phi \rangle) - F(\langle X_s^{(n)}, \phi \rangle) - \int_s^t \mathcal{A}_n F_\phi(X_u^{(n)}) du \right) \prod_{i=1}^k \Phi_i(X_{t_i}^{(n)}) \right] = 0. \quad (86) $$

Using a Taylor expansion, we have for some $\theta \in (0, 1)$:

$$
\begin{aligned}
F(\langle X_u^{(n)}, \phi \rangle - \langle \partial_2 \tau, \phi \rangle) &= F(\langle X_u^{(n)}, \phi \rangle) - \langle \partial_2 \tau, \phi \rangle F'(\langle X_u^{(n)}, \phi \rangle) \\
&\quad + \frac{1}{2} \langle \partial_2 \tau, \phi \rangle^2 F''(\langle X_u^{(n)}, \phi \rangle) + \frac{1}{6} \langle \partial_2 \tau, \phi \rangle^3 F^{(3)}(\langle X_u^{(n)}, \phi \rangle - \theta \langle \partial_2 \tau, \phi \rangle).
\end{aligned}
\quad (87)
$$

As a result,

$$
\begin{aligned}
& \left| \int_s^t \mathcal{A}_n F_\phi(X_u^{(n)}) du - \int_s^t \mathfrak{A} F_\phi(X_u^{(n)}) du \right| \\
&= \left| \epsilon_n^{-2} \int_s^t \sum_{\tau \in S_2^n} (F_\phi(X_u^{(n)} - \partial_2 \tau) - F_\phi(X_u^{(n)})) w(X_u^{(n)}, \partial_2 \tau) \, du - \int_s^t F'(\langle X_u^{(n)}, \phi \rangle) \int_{X_u^{(n)}} \mathcal{L}^\uparrow \phi \, du \right. \\
&\qquad \left. - \lim_{n \to +\infty} \epsilon_n^{-2} \int_s^t F''\left(\int_{X_u^{(n)}} \phi \right) \sum_{\tau \in S_2^n} \langle \partial_2 \tau, \phi \rangle^2 w(X_u^{(n)}, \partial_2 \tau) \, du \right| \\
&\le \int_s^t \left| F'_\phi(X_u^{(n)}) (\mathcal{A}_n(\langle X_u^{(n)}, \phi \rangle) - \langle X_u^{(n)}, \mathcal{L}^\uparrow \phi \rangle) \right| \, du \\
&\quad + \left| \frac{1}{2} \epsilon_n^{-2} \int_s^t F''(\langle X_u^{(n)}, \phi \rangle) \sum_{\tau \in S_2^n} \langle \partial_2 \tau, \phi \rangle^2 w(X_u^{(n)}, \partial_2 \tau) \, du \right| \\
&\quad - \lim_{n \to +\infty} \epsilon_n^{-2} \int_s^t F''\left(\int_{X_u^{(n)}} \phi \right) \sum_{\tau \in S_2^n} \langle \partial_2 \tau, \phi \rangle^2 w(X_u^{(n)}, \partial_2 \tau) \, du \right| \\
&\le C \|F'\|_{\infty} \epsilon_n \|\sigma\|_F + O(1) + O(\epsilon_n^2 \|\sigma\|_F),
\end{aligned}
$$

which converge in expectation to zero when $n \to +\infty$ under Conjecture 33. Then, the announced result is deduced because the functions $\Phi_i$ are bounded. ■

Let us now conclude. In view of Theorem 8.2 in [17, p.226], given the result of Proposition 44, every limiting process $\tilde{X}$ of the sequence $(X^n)$ is a solution of the martingale problem associated with $\mathfrak{A}$. If there is a unique solution to this martingale problem, then the whole sequence converges to this solution. The uniqueness problem is left open. This concludes the proof of Theorem 36.
**Corollary 43.** For $X_u \in \mathbb{C}_{1,4}$, recall the definition of $\tilde{X}_u$ in (59). We have:

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

**Proposition 44.** Let us consider $\phi \in \mathbb{C}^{1,4}$ and $F \in C_b^3(\mathbb{R})$ and recall the definition of $\mathfrak{A}$ in (61). Then, for $k \in \mathbb{N}$, for $0 \le t_1 \le \dots t_k \le s < t$, and $\Phi_1, \dots, \Phi_k$ bounded continuous functions of $\mathbb{C}_{1,4}$,

$$
\lim_{n \to +\infty} \mathbb{E} \left[ \left( F(\langle X_t^{(n)}, \phi \rangle) - F(\langle X_s^{(n)}, \phi \rangle) - \int_s^t \mathfrak{A} F_\phi(X_u^{(n)}) du \right) \prod_{i=1}^k \Phi_i(X_{t_i}^{(n)}) \right] = 0. \quad (85)
$$

*Proof.* Let us consider $s < t$. Using (50) and It's formula [26, Th.5.1 P.66] for jump processes, we have that for every $n \in \mathbb{N}$:

$$
\mathbb{E} \left[ \left( F(\langle X_t^{(n)}, \phi \rangle) - F(\langle X_s^{(n)}, \phi \rangle) - \int_s^t \mathcal{A}_n F_\phi(X_u^{(n)}) du \right) \prod_{i=1}^k \Phi_i(X_{t_i}^{(n)}) \right] = 0. \quad (86)
$$

Using a Taylor expansion, we have for some $\theta \in (0, 1)$:

$$
\begin{align}
F(\langle X_u^{(n)}, \phi \rangle - \langle \partial_2 \tau, \phi \rangle) &= F(\langle X_u^{(n)}, \phi \rangle) - \langle \partial_2 \tau, \phi \rangle F'(\langle X_u^{(n)}, \phi \rangle) \nonumber \\
&\quad + \frac{1}{2} \langle \partial_2 \tau, \phi \rangle^2 F''(\langle X_u^{(n)}, \phi \rangle) + \frac{1}{6} \langle \partial_2 \tau, \phi \rangle^3 F^{(3)}(\langle X_u^{(n)}, \phi \rangle - \theta \langle \partial_2 \tau, \phi \rangle). \tag{87}
\end{align}
$$

As a result,

$$
\begin{align*}
& \left| \int_s^t \mathcal{A}_n F_\phi(X_u^{(n)}) du - \int_s^t \mathfrak{A} F_\phi(X_u^{(n)}) du \right| \\
&= \left| \epsilon_n^{-2} \int_s^t \sum_{\tau \in S_2^n} (F_\phi(X_u^{(n)} - \partial_2 \tau) - F_\phi(X_u^{(n)})) w(X_u^{(n)}, \partial_2 \tau) \, du - \int_s^t F'(\langle X_u^{(n)}, \phi \rangle) \int_{X_u^{(n)}} \mathcal{L}^\uparrow \phi \, du \right. \\
&\qquad \left. - \lim_{n \to +\infty} \epsilon_n^{-2} \int_s^t F''\left(\int_{X_u^{(n)}} \phi \right) \sum_{\tau \in S_2^n} \langle \partial_2 \tau, \phi \rangle^2 w(X_u^{(n)}, \partial_2 \tau) \, du \right| \\
&\le \int_s^t \left| F'_\phi(X_u^{(n)}) (\mathcal{A}_n(\langle X_u^{(n)}, \phi \rangle) - \langle X_u^{(n)}, \mathcal{L}^\uparrow \phi \rangle) \right| du \\
&\quad + \left| \frac{1}{2} \epsilon_n^{-2} \int_s^t F''(\langle X_u^{(n)}, \phi \rangle) \sum_{\tau \in S_2^n} \langle \partial_2 \tau, \phi \rangle^2 w(X_u^{(n)}, \partial_2 \tau) \, du \right| \\
&\quad - \lim_{n \to +\infty} \epsilon_n^{-2} \int_s^t F''\left(\int_{X_u^{(n)}} \phi \right) \sum_{\tau \in S_2^n} \langle \partial_2 \tau, \phi \rangle^2 w(X_u^{(n)}, \partial_2 \tau) \, du \right| \\
&\le C \|F'\|_{\infty} \epsilon_n \|\sigma\|_F + O(1) + O(\epsilon_n^2 \|\sigma\|_F),
\end{align*}
$$

which converge in expectation to zero when $n \to +\infty$ under Conjecture 33. Then, the announced result is deduced because the functions $\Phi_i$ are bounded. ■

Let us now conclude. In view of Theorem 8.2 in [17, p.226], given the result of Proposition 44, every limiting process $\tilde{X}$ of the sequence $(X^n)$ is a solution of the martingale problem associated with $\mathfrak{A}$. If there is a unique solution to this martingale problem, then the whole sequence converges to this solution. The uniqueness problem is left open. This concludes the proof of Theorem 36.
# 5 Application

A straightforward application of our random walk is the enumeration of the elements of $ker L_k$ (i.e. chains which are cycles) with information only on the simplexes (i.e. the basis of $C_k$). Start with any cycle and let the random walk run to obtain a partial list of the chains which are in the homology class of the initial chain. The longest the run, the longest the list.

An avatar of this simple algorithm is to use the Markov chain in the context of the simulated annealing algorithm. This heuristic is devised to minimize an energy function through a random walk in the state space trying to avoid local minimums. We here focus on 1-chains but the procedure could be applied as is to higher order simplexes.

As we have seen above, the dimension of the kernel of $L_1$ is the Betti number $\beta_1$. Furthermore, any element of $ker L_1$ can be seen geometrically as a closed path (i.e. a cycle) which surrounds one or several holes, see Figure 8.

<figure><img src="image_12.png" /><figcaption>Figure 8: A simplicial complex with $\beta_1 = 2$ and a cycle which surrounds the two holes.</figcaption></figure>

It is important for some applications to be able to precise the holes locations, in order for instance to refill them (see e.g.[52]). The first idea which comes to mind is to find a closed path around each hole with minimum length. The algebraic representation of the geometric problem does not contain any information about the Euclidean length of a path so this definition of length cannot be used here. We thus have to rely on the algebraic length of a path, namely, the sum of the weight of each edge in a chain:

$$U(\sigma) = \sum_{\tau \in S_1^+} |\langle \tau^*, \sigma \rangle_{C^1, C_1}|.$$

For reasons which will be explained below, it is not always possible to define $U(\sigma)$ as the number of edges in $\sigma$. We here have

**Theorem 45.** For $k \ge 0$, let us consider the vector

$$\omega(t) = (\mathbf{P}(\langle \tau^*, X_t \rangle_{C^k, C_k} = 1 | X_0), \tau \in S_k).$$

The $C_k$-valued process $(\omega(t), t \ge 0)$ satisfies the differential equation

$$\frac{d}{dt} \omega(t) = -L_k^\uparrow \omega(t). \qquad (88)$$
# 5 Application

A straightforward application of our random walk is the enumeration of the elements of $ker L_k$ (i.e. chains which are cycles) with information only on the simplexes (i.e. the basis of $C_k$). Start with any cycle and let the random walk run to obtain a partial list of the chains which are in the homology class of the initial chain. The longest the run, the longest the list.

An avatar of this simple algorithm is to use the Markov chain in the context of the simulated annealing algorithm. This heuristic is devised to minimize an energy function through a random walk in the state space trying to avoid local minimums. We here focus on 1-chains but the procedure could be applied as is to higher order simplexes.

As we have seen above, the dimension of the kernel of $L_1$ is the Betti number $\beta_1$. Furthermore, any element of $ker L_1$ can be seen geometrically as a closed path (i.e. a cycle) which surrounds one or several holes, see Figure 8.

<figure><img src="image_15.png" /><figcaption>Figure 8: A simplicial complex with $\beta_1 = 2$ and a cycle which surrounds the two holes.</figcaption></figure>

It is important for some applications to be able to precise the holes locations, in order for instance to refill them (see e.g.[52]). The first idea which comes to mind is to find a closed path around each hole with minimum length. The algebraic representation of the geometric problem does not contain any information about the Euclidean length of a path so this definition of length cannot be used here. We thus have to rely on the algebraic length of a path, namely, the sum of the weight of each edge in a chain:

$$U(\sigma) = \sum_{\tau \in S_1^+} |\langle \tau^*, \sigma \rangle_{C^1, C_1}|.$$

For reasons which will be explained below, it is not always possible to define $U(\sigma)$ as the number of edges in $\sigma$. We here have

**Theorem 45.** For $k \ge 0$, let us consider the vector

$$\omega(t) = \left( \mathbf{P} \left( \langle \tau^*, X_t \rangle_{C^k, C_k} = 1 | X_0 \right), \tau \in S_k \right).$$

The $C_k$-valued process $(\omega(t), t \ge 0)$ satisfies the differential equation

$$\frac{d}{dt} \omega(t) = -L_k^\uparrow \omega(t). \qquad (88)$$
It is shown in [38] that the solution of the differential equation

$$ \frac{d}{dt} \omega(t) = -L_k \omega(t). \qquad (89) $$

converges as $t$ goes to infinity towards $\omega_\infty \in \ker L_k$. An element of $\ker L_k$ is a cycle and the numerical experiments of [38] tend to show that the weights of $\omega_\infty$ are heavier around the hole.

*Proof.* For $\tau$ a given $k$-simplex, consider the function

$$ \Theta_{\tau} : C_k \to \mathbf{R} \\ \sigma \mapsto \langle \tau^*, \sigma \rangle_{C^k, C_k}^{+} $$

If $\sigma$ is a simple chain, $\Theta_\tau(\sigma) = 1$ whenever $\tau$ (taking care of the orientation) belongs to the support of $\sigma$ and 0 otherwise, i.e.

$$ \Theta_{\tau}(\sigma) = \mathbf{1}_{\text{supp } \sigma}(\tau). $$

The function $\Theta_\tau$ is not a linear function but we can proceed as in the proof of Theorem 18 to establish that

$$ A\Theta_{\tau}(\sigma) = -L_k^{\uparrow}\Theta_{\tau}(\sigma) $$

by noticing that for $\sigma' \sim \sigma$, $\sigma'$ differs from $\sigma$ only by the boundary of a $(k+1)$-simplex and hence:

$$ \Theta_{\tau}(\sigma) - \Theta_{\tau}(\sigma') = \Theta_{\tau}(\sigma - \sigma'). $$

We conclude by the Chapman-Kolmogorov equations for continuous time Markov chains.

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

Then, our first hope was that by letting the random walk run, the most visited edges would be located around the holes. Our numerical experiments showed disappointing results probably because of a too slow convergence.

The determination of a shortest path around some hole has strong reminiscence with the so-called minimum cycle bases in graphs (see [35] and references therein) but the algorithms developed there cannot be used here since they, by essence, cannot take into account the topological features like holes. For instance, the Horton's collection algorithm starts by computing all the shortest cycles $C(v, e)$ which start at a vertex $v$ and contains an edge $e$ containing $v$. Then, it computes some linear combinations of these cycles to find a minimum weight bases. We immediately see that there is no guarantee that these linear combinations are in the convenient homology class since this information is not contained in $\{C(v, e), v \in S_0, e \in S_1\}$. Moreover, the number of paths is usually so huge that even a polynomial algorithm in the number of paths would be unrealistic. We here resort to a simulated annealing (SA for short) algorithm where the energy function to be minimized is simply $U$ as defined above. There is no guarantee that the SA algorithm is not stuck in a local minimum of $U$ but in practice, it works pretty well.

We initiate the SA algorithm with an element $\tau_0$ of $\ker L_1$ and a temperature $T_0$. We fix some $\alpha \in (0, 1)$. Assume that the SA algorithm has already made $m$ steps with $m = 0, \dots$. The new temperature is $T_{m+1} = T_0 \alpha^{m+1}$. We apply a transition of our Markov chain from the chain $\sigma_m$ and get a chain $\sigma'$. If $U(\sigma') < U(\sigma_m)$ then $\sigma_{m+1} = \sigma'$. If $U(\sigma') > U(\sigma_m)$, the new state is $\sigma'$ with probability

$$ \exp\left(-\frac{U(\sigma') - U(\sigma_m)}{T_{m+1}}\right) $$

and $\sigma_{m+1} = \sigma_m$ with the complementary probability. We stop the algorithm when the temperature is below a predefined threshold and we expect that at this time, the state of the algorithm yields a short path around one or several holes.

In the situation of Figure 8, we know easily a path (i.e. a chain with unit weights) around the two holes. After a few iterations of the SA algorithm, we obtain a very neat answer to our problem illustrated in Figure 9.

It is not always feasible to describe simply a path around the holes. Then, we can use as $\sigma_0$ an element of $\ker L_1$ obtained from the Smith normal form of $L_1$ (we must be careful not to work with
It is shown in [38] that the solution of the differential equation

$$ \frac{d}{dt}\omega(t) = -L_k\omega(t). \qquad (89) $$

converges as $t$ goes to infinity towards $\omega_\infty \in \ker L_k$. An element of $\ker L_k$ is a cycle and the numerical experiments of [38] tend to show that the weights of $\omega_\infty$ are heavier around the hole.

*Proof.* For $\tau$ a given $k$-simplex, consider the function

$$ \Theta_{\tau} : C_k \to \mathbf{R} \\ \sigma \mapsto \langle \tau^*, \sigma \rangle_{C^k, C_k}^{+}. $$

If $\sigma$ is a simple chain, $\Theta_{\tau}(\sigma) = 1$ whenever $\tau$ (taking care of the orientation) belongs to the support of $\sigma$ and 0 otherwise, i.e.

$$ \Theta_{\tau}(\sigma) = \mathbf{1}_{\text{supp } \sigma}(\tau). $$

The function $\Theta_{\tau}$ is not a linear function but we can proceed as in the proof of Theorem 18 to establish that

$$ A\Theta_{\tau}(\sigma) = -L_k^{\uparrow}\Theta_{\tau}(\sigma) $$

by noticing that for $\sigma' \sim \sigma$, $\sigma'$ differs from $\sigma$ only by the boundary of a $(k+1)$-simplex and hence:

$$ \Theta_{\tau}(\sigma) - \Theta_{\tau}(\sigma') = \Theta_{\tau}(\sigma - \sigma'). $$

We conclude by the Chapman-Kolmogorov equations for continuous time Markov chains.

<figure><img src="image_16.png" /></figure>

Then, our first hope was that by letting the random walk run, the most visited edges would be located around the holes. Our numerical experiments showed disappointing results probably because of a too slow convergence.

The determination of a shortest path around some hole has strong reminiscence with the so-called minimum cycle bases in graphs (see [35] and references therein) but the algorithms developed there cannot be used here since they, by essence, cannot take into account the topological features like holes. For instance, the Horton's collection algorithm starts by computing all the shortest cycles $C(v, e)$ which start at a vertex $v$ and contains an edge $e$ containing $v$. Then, it computes some linear combinations of these cycles to find a minimum weight bases. We immediately see that there is no guarantee that these linear combinations are in the convenient homology class since this information is not contained in $\{C(v, e), v \in S_0, e \in S_1\}$. Moreover, the number of paths is usually so huge that even a polynomial algorithm in the number of paths would be unrealistic. We here resort to a simulated annealing (SA for short) algorithm where the energy function to be minimized is simply $U$ as defined above. There is no guarantee that the SA algorithm is not stuck in a local minimum of $U$ but in practice, it works pretty well.

We initiate the SA algorithm with an element $\tau_0$ of $\ker L_1$ and a temperature $T_0$. We fix some $\alpha \in (0, 1)$. Assume that the SA algorithm has already made $m$ steps with $m = 0, \dots$. The new temperature is $T_{m+1} = T_0\alpha^{m+1}$. We apply a transition of our Markov chain from the chain $\sigma_m$ and get a chain $\sigma'$. If $U(\sigma') < U(\sigma_m)$ then $\sigma_{m+1} = \sigma'$. If $U(\sigma') > U(\sigma_m)$, the new state is $\sigma'$ with probability

$$ \exp\left(-\frac{U(\sigma') - U(\sigma_m)}{T_{m+1}}\right) $$

and $\sigma_{m+1} = \sigma_m$ with the complementary probability. We stop the algorithm when the temperature is below a predefined threshold and we expect that at this time, the state of the algorithm yields a short path around one or several holes.

In the situation of Figure 8, we know easily a path (i.e. a chain with unit weights) around the two holes. After a few iterations of the SA algorithm, we obtain a very neat answer to our problem illustrated in Figure 9.

It is not always feasible to describe simply a path around the holes. Then, we can use as $\sigma_0$ an element of $\ker L_1$ obtained from the Smith normal form of $L_1$ (we must be careful not to work with
<figure><img src="image_14.png" /><figcaption>Figure 9: The result of the SA algorithm with a manually defined initial chain.</figcaption></figure>

floating numbers since the computations error quickly propagate and the final result is no longer a cycle). The price to pay for such a generality is that the weights of the components of $\sigma_0$ are usually very large integers and almost no edge has a zero weight, see the left picture of Figure 10. After the SA algorithm has been run, the edges around the hole have a much greater weight meanwhile the other edges see their weight almost unchanged or even decreased. It remains to remove the uninteresting edges by a cut-off procedure to have a good location of the hole.

<figure><img src="image_15.png" /><figcaption>Figure 10: Another simplicial complex with $\beta_1 = 1$. The width of the edge and the darkness of the blue color are proportional to the weight of the edge given by the Smith normal form of $L_1$. On the left, some edges not adjacent to the hole have a relative great weight. After the SA has been run, these weights are no longer so important and the edges around the hole are much heavier.</figcaption></figure>

According to the results of [10, 50], one can show (see [60]) that there exists an optimal cooling scheme, i.e. an optimal $\alpha$, such that

$$ \sup_{\sigma_0 \in \ker L_1} \mathbf{P}_{\sigma_0}(U(\sigma_m) > \min U) \le \frac{c}{m^{1/L}} $$

where $L = \max\{U(\sigma), \sigma$ in the homology class of $\sigma_0\}$. On the examples we test our SA algorithm with, the localisation was obtained much faster.
<figure><img src="image_17.png" /><figcaption>Figure 9: The result of the SA algorithm with a manually defined initial chain.</figcaption></figure>

floating numbers since the computations error quickly propagate and the final result is no longer a cycle). The price to pay for such a generality is that the weights of the components of $\sigma_0$ are usually very large integers and almost no edge has a zero weight, see the left picture of Figure 10. After the SA algorithm has been run, the edges around the hole have a much greater weight meanwhile the other edges see their weight almost unchanged or even decreased. It remains to remove the uninteresting edges by a cut-off procedure to have a good location of the hole.

<figure><img src="image_18.png" /><figcaption>Figure 10: Another simplicial complex with $\beta_1 = 1$. The width of the edge and the darkness of the blue color are proportional to the weight of the edge given by the Smith normal form of $L_1$. On the left, some edges not adjacent to the hole have a relative great weight. After the SA has been run, these weights are no longer so important and the edges around the hole are much heavier.</figcaption></figure>

According to the results of [10, 50], one can show (see [60]) that there exists an optimal cooling scheme, i.e. an optimal $\alpha$, such that

$$ \sup_{\sigma_0 \in \ker L_1} \mathbf{P}_{\sigma_0}(U(\sigma_m) > \min U) \le \frac{c}{m^{1/L}} $$

where $L = \max\{U(\sigma), \sigma$ in the homology class of $\sigma_0\}$. On the examples we test our SA algorithm with, the localisation was obtained much faster.
A technical Lemma for integrating a function $H(x_1, x_2)$ on small triangles of the torus

**Lemma 46.** Let $H: \mathbb{T}_2 \to \mathbb{R}$ be $\mathbb{C}^3$. Let us consider a triangle of $\mathcal{T}_n$. Without loss of generality, we can assume (by a change of coordinates) that it is the triangle [021] whose vertices have coordinate $(0,0)$, $(\epsilon_n, \sqrt{3}\epsilon_n)$ and $(2\epsilon_n, 0)$. Then, as $\epsilon_n$ goes to 0,

$$ 
\begin{aligned}
\int_{[021]} H(x_1, x_2) \, dx_2 \, dx_1 &= -\sqrt{3}H(0,0)\epsilon_n^2 - (\sqrt{3}H_1(0,0) + H_2(0,0))\epsilon_n^3 \\
&\quad -\left(\frac{7\sqrt{3}}{6}H_{11}(0,0) + \frac{\sqrt{3}}{2}H_{22}(0,0) + 2H_{12}(0,0)\right)\epsilon_n^4 + O(\epsilon_n^5) \\
&= -\frac{\sqrt{3}}{2}\epsilon_n \int_0^{2\epsilon_n} H(x_1, 0) \, dx_1 + O(\epsilon_n^3).
\end{aligned} 
\quad (90) $$

*Proof.* Using Fubini's theorem and taking into account the orientations,

$$ 
\begin{aligned}
M_1 &:= \int_{[021]} H(x_1, x_2) \, dx_2 \, dx_1 \\
&= -\int_0^{\epsilon_n} \int_0^{x_1\sqrt{3}} H(x_1, x_2) \, dx_2 \, dx_1 - \int_{\epsilon_n}^{2\epsilon_n} \int_0^{\sqrt{3}(2\epsilon_n - x_1)} H(x_1, x_2) \, dx_2 \, dx_1, \\
&=: -I - II.
\end{aligned} 
$$

A Taylor expansion gives

$$ H(x_1, x_2) = H(0,0) + x_1 H_1(0,0) + x_2 H_2(0,0) + x_1^2 H_{11}(0,0) + x_2^2 H_{22}(0,0) + 2x_1x_2H_{12}(0,0) \\ + x_1^3 r_{111}(x_1, x_2) + x_1^2 x_2 r_{112}(x_1, x_2) + x_1 x_2^2 r_{122}(x_1, x_2) + x_2^3 r_{222}(x_1, x_2) \quad (91) $$

where the remainder terms $r_{111}$, $r_{112}$, $r_{122}$ and $r_{222}$ satisfy:

$$ \sup_{x_1, x_2 \in [0,1]^2} (|r_{111}(x_1, x_2)|, |r_{122}(x_1, x_2)|, |r_{122}(x_1, x_2)|, |r_{222}(x_1, x_2)|) < +\infty, $$

with a bound that we can chose to depend only on $H$. Then, injecting (91) in $I$,

$$ 
\begin{aligned}
I &= H(0,0) \int_0^{\epsilon_n} x_2 \sqrt{3} \, dx_1 + H_1(0,0) \int_0^{\epsilon_n} x_1^2 \sqrt{3} \, dx_1 + H_2(0,0) \int_0^{\epsilon_n} \frac{3x_1^2}{2} \, dx_1 \\
&\quad + H_{11}(0,0) \int_0^{\epsilon_n} x_1^3 \sqrt{3} \, dx_1 + H_{22}(0,0) \int_0^{\epsilon_n} x_1^3 \sqrt{3} \, dx_1 + 2H_{12}(0,0) \int_0^{\epsilon_n} x_1 \int_0^{x_1\sqrt{3}} x_2 \, dx_2 \, dx_1 + R_n^I, \\
&= \frac{\sqrt{3}}{2} H(0,0) \epsilon_n^2 + \left( \frac{\sqrt{3}}{3} H_1(0,0) + \frac{1}{2} H_2(0,0) \right) \epsilon_n^3 + \left( \frac{\sqrt{3}}{4} H_{11}(0,0) + \frac{\sqrt{3}}{4} H_{22}(0,0) + \frac{3}{4} H_{12}(0,0) \right) \epsilon_n^4 + R_n^I, \\
\end{aligned} 
\quad (92) $$

where the remainder term $R_n^I$ satisfies $R_n^I = O(\epsilon_n^5)$. A similar computation shows that:

$$ 
\begin{aligned}
II &= \frac{\sqrt{3}}{2} H(0,0) \epsilon_n^2 + \left( \frac{2\sqrt{3}}{3} H_1(0,0) + \frac{1}{2} H_2(0,0) \right) \epsilon_n^3 \\
&\quad + \left( \frac{11\sqrt{3}}{12} H_{11}(0,0) + \frac{\sqrt{3}}{4} H_{22}(0,0) + \frac{5}{4} H_{12}(0,0) \right) \epsilon_n^4 + R_n^{II},
\end{aligned} 
\quad (93) $$

with $R_n^{II} = O(\epsilon_n^5)$. The results stem from the addition of (92) and (93).

<figure><img src="image_16.png" /></figure>
A technical Lemma for integrating a function $H(x_1, x_2)$ on small triangles of the torus

**Lemma 46.** Let $H: \mathbb{T}_2 \to \mathbb{R}$ be $\mathbb{C}^3$. Let us consider a triangle of $\mathcal{T}_n$. Without loss of generality, we can assume (by a change of coordinates) that it is the triangle [021] whose vertices have coordinate $(0,0)$, $(\epsilon_n, \sqrt{3}\epsilon_n)$ and $(2\epsilon_n, 0)$. Then, as $\epsilon_n$ goes to 0,

$$ 
\begin{aligned}
\int_{[021]} H(x_1, x_2) \, dx_2 \, dx_1 &= -\sqrt{3}H(0,0)\epsilon_n^2 - (\sqrt{3}H_1(0,0) + H_2(0,0))\epsilon_n^3 \\
&\quad - \left(\frac{7\sqrt{3}}{6}H_{11}(0,0) + \frac{\sqrt{3}}{2}H_{22}(0,0) + 2H_{12}(0,0)\right)\epsilon_n^4 + O(\epsilon_n^5) \\
&= -\frac{\sqrt{3}}{2}\epsilon_n \int_0^{2\epsilon_n} H(x_1, 0) \, dx_1 + O(\epsilon_n^3).
\end{aligned}
\quad (90) $$

*Proof.* Using Fubini's theorem and taking into account the orientations,

$$ 
\begin{aligned}
M_1 &:= \int_{[021]} H(x_1, x_2) \, dx_2 \, dx_1 \\
&= - \int_0^{\epsilon_n} \int_0^{x_1\sqrt{3}} H(x_1, x_2) \, dx_2 \, dx_1 - \int_{\epsilon_n}^{2\epsilon_n} \int_0^{\sqrt{3}(2\epsilon_n - x_1)} H(x_1, x_2) \, dx_2 \, dx_1, \\
&=: -I - II.
\end{aligned}
$$

A Taylor expansion gives

$$ H(x_1, x_2) = H(0,0) + x_1 H_1(0,0) + x_2 H_2(0,0) + x_1^2 H_{11}(0,0) + x_2^2 H_{22}(0,0) + 2x_1 x_2 H_{12}(0,0) \\ + x_1^3 r_{111}(x_1, x_2) + x_1^2 x_2 r_{112}(x_1, x_2) + x_1 x_2^2 r_{122}(x_1, x_2) + x_2^3 r_{222}(x_1, x_2) \quad (91) $$

where the remainder terms $r_{111}$, $r_{112}$, $r_{122}$ and $r_{222}$ satisfy:

$$ \sup_{x_1, x_2 \in [0,1]^2} (|r_{111}(x_1, x_2)|, |r_{122}(x_1, x_2)|, |r_{122}(x_1, x_2)|, |r_{222}(x_1, x_2)|) < +\infty, $$

with a bound that we can chose to depend only on $H$. Then, injecting (91) in $I$,

$$ 
\begin{aligned}
I &= H(0,0) \int_0^{\epsilon_n} x_2 \sqrt{3} \, dx_1 + H_1(0,0) \int_0^{\epsilon_n} x_1^2 \sqrt{3} \, dx_1 + H_2(0,0) \int_0^{\epsilon_n} \frac{3x_1^2}{2} \, dx_1 \\
&\quad + H_{11}(0,0) \int_0^{\epsilon_n} x_1^3 \sqrt{3} \, dx_1 + H_{22}(0,0) \int_0^{\epsilon_n} x_1^3 \sqrt{3} \, dx_1 + 2H_{12}(0,0) \int_0^{\epsilon_n} x_1 \int_0^{x_1\sqrt{3}} x_2 \, dx_2 \, dx_1 + R_n^I, \\
&= \frac{\sqrt{3}}{2} H(0,0) \epsilon_n^2 + \left( \frac{\sqrt{3}}{3} H_1(0,0) + \frac{1}{2} H_2(0,0) \right) \epsilon_n^3 + \left( \frac{\sqrt{3}}{4} H_{11}(0,0) + \frac{\sqrt{3}}{4} H_{22}(0,0) + \frac{3}{4} H_{12}(0,0) \right) \epsilon_n^4 + R_n^I, \\
\end{aligned}
\quad (92) $$

where the remainder term $R_n^I$ satisfies $R_n^I = O(\epsilon_n^5)$. A similar computation shows that:

$$ 
\begin{aligned}
II &= \frac{\sqrt{3}}{2} H(0,0) \epsilon_n^2 + \left( \frac{2\sqrt{3}}{3} H_1(0,0) + \frac{1}{2} H_2(0,0) \right) \epsilon_n^3 \\
&\quad + \left( \frac{11\sqrt{3}}{12} H_{11}(0,0) + \frac{\sqrt{3}}{4} H_{22}(0,0) + \frac{5}{4} H_{12}(0,0) \right) \epsilon_n^4 + R_n^{II},
\end{aligned}
\quad (93) $$

with $R_n^{II} = O(\epsilon_n^5)$. The results stems from the addition of (92) and (93). ■
## B A technical lemma for integrating the function $c$ and its derivatives on small triangles of the torus

**Lemma 47.** Consider the triangle [021] (with the notation of Fig. 6) whose barycenter is $G = (\epsilon_n, \epsilon_n/\sqrt{3})$. We have:

$$ 
\begin{align} 
\int_{[021]} c(x_1, x_2)dx_1 dx_2 &= \frac{5}{9} \epsilon_n^2 \sqrt{3}, \\ 
\int_{[021]} (x_1 - \epsilon_n)c(x_1, x_2)dx_1 dx_2 &= 0, \\ 
\int_{[021]} \frac{\partial c}{\partial x_1}(x_1, x_2)dx_1 dx_2 &= 0, \\ 
\int_{[021]} (x_1 - \epsilon_n)\frac{\partial c}{\partial x_1}(x_1, x_2)dx_1 dx_2 &= -\frac{2}{9}\epsilon_n^2 \sqrt{3}, \tag{94}
\end{align} 
$$

$$ \int_{[021]} \left(x_2 - \frac{\sqrt{3}}{3}\epsilon_n\right) \frac{\partial c}{\partial x_1}(x_1, x_2)dx_1 dx_2 = 0. \quad (95) $$

*Proof.* Let us denote by $C_1$, $C_2$ and $C_3$ the Voronoi cells associated with the three vertices (see Fig. 11).

<figure><img src="image_17.png" /><figcaption>Figure 11: Triangle [012]. $0 := (0,0)$, $1 := (2\epsilon_n, 0)$, $2 := (\epsilon_n, \sqrt{3}\epsilon_n)$, $3 := (\epsilon_n, -\sqrt{3}\epsilon_n)$.</figcaption></figure>

We have:

$$ c(x_1, x_2) = \frac{1}{\epsilon_n^2} \times \begin{cases} x_1^2 + x_2^2 & \text{on } C_1 \\ (x_1 - 2\epsilon_n)^2 + x_2^2 & \text{on } C_2 \\ (x_1 - \epsilon_n)^2 + (x_2 - \epsilon_n\sqrt{3})^2 & \text{on } C_3. \end{cases} $$

and,

$$ \frac{\partial c}{\partial x_1}(x_1, x_2) = \frac{1}{\epsilon_n^2} \times \begin{cases} 2x_1 & \text{on } C_1 \\ 2(x_1 - 2\epsilon_n) & \text{on } C_2 \\ 2(x_1 - \epsilon_n) & \text{on } C_3. \end{cases} $$

By symmetry,

$$ 
\begin{align} 
\int_{[021]} c(x_1, x_2)dx_2 dx_1 &= 6 \times \int_0^{\epsilon_n} \int_0^{x_1/\sqrt{3}} \frac{x_1^2 + x_2^2}{\epsilon_n^2} dx_2 dx_1 \nonumber \\ 
&= \frac{6}{\epsilon_n^2} \int_0^{\epsilon_n} \left( \frac{x_1^3}{\sqrt{3}} + \frac{x_1^3}{9\sqrt{3}} \right) dx_1 \nonumber \\ 
&= \frac{6}{\epsilon_n^2} \frac{10}{9\sqrt{3}} \frac{\epsilon_n^4}{4} = \frac{5}{9}\epsilon_n^2 \sqrt{3}. \tag{96}
\end{align} 
$$
B A technical lemma for integrating the function $c$ and its
derivatives on small triangles of the torus

**Lemma 47.** Consider the triangle [021] (with the notation of Fig. 6) whose barycenter is $G = (\epsilon_n, \epsilon_n/\sqrt{3})$. We have:

$$
\begin{align}
\int_{[021]} c(x_1, x_2)dx_1 dx_2 &= \frac{5}{9} \epsilon_n^2 \sqrt{3}, \nonumber \\
\int_{[021]} (x_1 - \epsilon_n)c(x_1, x_2)dx_1 dx_2 &= 0, \nonumber \\
\int_{[021]} \frac{\partial c}{\partial x_1}(x_1, x_2)dx_1 dx_2 &= 0, \nonumber \\
\int_{[021]} (x_1 - \epsilon_n) \frac{\partial c}{\partial x_1}(x_1, x_2)dx_1 dx_2 &= -\frac{2}{9} \epsilon_n^2 \sqrt{3}, \tag{94}
\end{align}
$$

$$
\int_{[021]} \left(x_2 - \frac{\sqrt{3}}{3}\epsilon_n\right) \frac{\partial c}{\partial x_1}(x_1, x_2)dx_1 dx_2 = 0. \quad (95)
$$

*Proof.* Let us denote by $C_1$, $C_2$ and $C_3$ the Voronoi cells associated with the three vertices (see Fig. 11).

<figure><img src="image_19.png" /><figcaption>Figure 11: Triangle [012]. 0 := (0,0), 1 := (2epsilon_n, 0), 2 := (epsilon_n,sqrt3epsilon_n), 3 := (epsilon_n,-sqrt3epsilon_n).</figcaption></figure>

We have:

$$
c(x_1, x_2) = \frac{1}{\epsilon_n^2} \times \begin{cases} x_1^2 + x_2^2 & \text{on } C_1 \\ (x_1 - 2\epsilon_n)^2 + x_2^2 & \text{on } C_2 \\ (x_1 - \epsilon_n)^2 + (x_2 - \epsilon_n\sqrt{3})^2 & \text{on } C_3. \end{cases}
$$

and,

$$
\frac{\partial c}{\partial x_1}(x_1, x_2) = \frac{1}{\epsilon_n^2} \times \begin{cases} 2x_1 & \text{on } C_1 \\ 2(x_1 - 2\epsilon_n) & \text{on } C_2 \\ 2(x_1 - \epsilon_n) & \text{on } C_3. \end{cases}
$$

By symmetry,

$$
\begin{align}
\int_{[021]} c(x_1, x_2)dx_2dx_1 &= 6 \times \int_0^{\epsilon_n} \int_0^{x_1/\sqrt{3}} \frac{x_1^2 + x_2^2}{\epsilon_n^2} dx_2 dx_1 \nonumber \\
&= \frac{6}{\epsilon_n^2} \int_0^{\epsilon_n} \left( \frac{x_1^3}{\sqrt{3}} + \frac{x_1^3}{9\sqrt{3}} \right) dx_1 \nonumber \\
&= \frac{6}{\epsilon_n^2} \frac{10}{9\sqrt{3}} \frac{\epsilon_n^4}{4} = \frac{5}{9} \epsilon_n^2 \sqrt{3}. \tag{96}
\end{align}
$$
Let us now compute the fourth equality. We have:

$$ \int_{[021]} (x_1 - \epsilon_n) \frac{\partial c}{\partial x_1}(x_1, x_2) dx_2 dx_1 = \sum_{i=1}^{3} \int_{C_i} (x_1 - \epsilon_n) \frac{\partial c}{\partial x_1}(x_1, x_2) dx_2 dx_1. \quad (97) $$

By symmetry, the integrals on the cells $C_1$ and $C_2$ are the same and:

$$ \begin{aligned} \int_{C_1} (x_1 - \epsilon_n) \frac{\partial c}{\partial x_1}(x_1, x_2) dx_2 dx_1 &= \int_0^{\epsilon_n/2} \int_0^{x_1\sqrt{3}} \frac{2x_1}{\epsilon_n^2} (x_1 - \epsilon_n) dx_2 dx_1 \\ &\quad + \int_{\epsilon_n/2}^{\epsilon_n} \int_0^{\epsilon_n\sqrt{3}/2 - (x_1 - \epsilon_n/2)/\sqrt{3}} \frac{2x_1}{\epsilon_n^2} (x_1 - \epsilon_n) dx_2 dx_1 \\ &= -\frac{1}{8} \epsilon_n^2 \sqrt{3}. \end{aligned} \quad (98) $$

For the cell $C_3$:

$$ \begin{aligned} \int_{C_3} (x_1 - \epsilon_n) \frac{\partial c}{\partial x_1}(x_1, x_2) dx_2 dx_1 &= \frac{2}{\epsilon_n^2} \int_{\epsilon_n/\sqrt{3}}^{\epsilon_n\sqrt{3}/2} \int_{\epsilon_n - \sqrt{3}(x_2 - \epsilon_n/\sqrt{3})}^{\epsilon_n + \sqrt{3}(x_2 - \epsilon_n/\sqrt{3})} (x_1 - \epsilon_n)^2 dx_1 dx_2 \\ &\quad + \frac{2}{\epsilon_n^2} \int_{\epsilon_n/\sqrt{3}/2}^{\epsilon_n/\sqrt{3}} \int_{\epsilon_n - \frac{1}{\sqrt{3}}(\epsilon_n\sqrt{3} - x_2)}^{\epsilon_n + \frac{1}{\sqrt{3}}(\epsilon_n\sqrt{3} - x_2)} (x_1 - \epsilon_n)^2 dx_1 dx_2 \\ &= \frac{2}{\epsilon_n^2} \int_{\epsilon_n/\sqrt{3}}^{\epsilon_n\sqrt{3}/2} \frac{2}{3} \times 3\sqrt{3} (x_2 - \frac{\epsilon_n\sqrt{3}}{3})^3 dx_2 + \frac{2}{\epsilon_n^2} \int_{\epsilon_n/\sqrt{3}/2}^{\epsilon_n/\sqrt{3}} \frac{2}{3} \left( \frac{1}{\sqrt{3}} (\epsilon_n\sqrt{3} - x_2) \right)^3 \quad (99) \\ &= \frac{\sqrt{3}}{\epsilon_n^2} \left( \frac{\epsilon_n\sqrt{3}}{2} - \frac{\epsilon_n\sqrt{3}}{3} \right)^4 + \frac{1}{9\sqrt{3}\epsilon_n^2} \frac{9\epsilon_n^4}{2^4} \\ &= \left( \frac{1}{9 \times 2^4} + \frac{1}{3 \times 2^4} \right) \epsilon_n^2 \sqrt{3} = \frac{1}{36} \epsilon_n^2 \sqrt{3}. \end{aligned} $$

Gathering (97), (96) and (99),

$$ \int_{[021]} (x_1 - \epsilon_n) \frac{\partial c}{\partial x_1}(x_1, x_2) dx_2 dx_1 = -\frac{2}{9} \epsilon_n^2 \sqrt{3}. \quad (100) $$

The second, third and fifth equalities of (94) stem from the fact that $G$ is the barycenter of the triangle. ■

## C Proof of Corollary 43

We can now prove Corollary 43 that is useful for the computation of the generator in Section 4.5.4:

*Proof of Corollary 43.* The first equality is just the definition of $\ast d\phi$, see (49).

Let us consider the second equality. For a cycle $\sigma \in \mathfrak{C}_{1,4}$, let us consider the term

$$ \begin{aligned} \epsilon_n^{-2} \sum_{\tau \in S_2^n} \langle \partial_2 \tau, \phi \rangle^2 w(\sigma, \partial_2 \tau) &= \epsilon_n^{-2} \sum_{e \in \sigma} (\langle \partial_2 \tau^+, \phi \rangle^2 + \langle \partial_2 \tau^-, \phi \rangle^2) \\ &= 2 \epsilon_n^2 \sum_{e \in \sigma} \langle \partial_2 \tau^+, \phi \rangle^2 \end{aligned} $$

where $\tau^+$ and $\tau^-$ are the triangles adjacent to the edge $e \in \sigma$. Without loss of generality, consider $e = [01]$ in the notation of Fig. 6. We deduce from (65) that:

$$ \langle \partial_2 \tau^+, \phi \rangle^2 = \frac{3}{2} \epsilon_n^3 \int_0^{2\epsilon_n} (\phi_1^2(x_1, 0) - \phi_2^1(x_1, 0))^2 dx_1 + O(\epsilon_n^5). $$
Let us now compute the fourth equality. We have:

$$ \int_{[021]} (x_1 - \epsilon_n) \frac{\partial c}{\partial x_1} (x_1, x_2) dx_2 dx_1 = \sum_{i=1}^{3} \int_{C_i} (x_1 - \epsilon_n) \frac{\partial c}{\partial x_1} (x_1, x_2) dx_2 dx_1. \quad (97) $$

By symmetry, the integrals on the cells $C_1$ and $C_2$ are the same and:

$$ \begin{aligned} \int_{C_1} (x_1 - \epsilon_n) \frac{\partial c}{\partial x_1} (x_1, x_2) dx_2 dx_1 &= \int_0^{\epsilon_n/2} \int_0^{x_1\sqrt{3}} \frac{2x_1}{\epsilon_n^2} (x_1 - \epsilon_n) dx_2 dx_1 \\ &\quad + \int_{\epsilon_n/2}^{\epsilon_n} \int_0^{\epsilon_n\sqrt{3}/2 - (x_1 - \epsilon_n/2)/\sqrt{3}} \frac{2x_1}{\epsilon_n^2} (x_1 - \epsilon_n) dx_2 dx_1 \\ &= -\frac{1}{8} \epsilon_n^2 \sqrt{3}. \end{aligned} \quad (98) $$

For the cell $C_3$:

$$ \begin{aligned} \int_{C_3} (x_1 - \epsilon_n) \frac{\partial c}{\partial x_1} (x_1, x_2) dx_2 dx_1 &= \frac{2}{\epsilon_n^2} \int_{\epsilon_n/\sqrt{3}}^{\epsilon_n\sqrt{3}/2} \int_{\epsilon_n - \sqrt{3}(x_2 - \epsilon_n/\sqrt{3})}^{\epsilon_n + \sqrt{3}(x_2 - \epsilon_n/\sqrt{3})} (x_1 - \epsilon_n)^2 dx_1 dx_2 \\ &\quad + \frac{2}{\epsilon_n^2} \int_{\epsilon_n\sqrt{3}/2}^{\epsilon_n\sqrt{3}} \int_{\epsilon_n - \frac{1}{\sqrt{3}}(\epsilon_n\sqrt{3} - x_2)}^{\epsilon_n + \frac{1}{\sqrt{3}}(\epsilon_n\sqrt{3} - x_2)} (x_1 - \epsilon_n)^2 dx_1 dx_2 \\ &= \frac{2}{\epsilon_n^2} \int_{\epsilon_n/\sqrt{3}}^{\epsilon_n\sqrt{3}/2} \frac{2}{3} \times 3\sqrt{3} (x_2 - \frac{\epsilon_n\sqrt{3}}{3})^3 dx_2 + \frac{2}{\epsilon_n^2} \int_{\epsilon_n\sqrt{3}/2}^{\epsilon_n\sqrt{3}} \frac{2}{3} \left( \frac{1}{\sqrt{3}} (\epsilon_n\sqrt{3} - x_2) \right)^3 \quad (99) \\ &= \frac{\sqrt{3}}{\epsilon_n^2} \left( \frac{\epsilon_n\sqrt{3}}{2} - \frac{\epsilon_n\sqrt{3}}{3} \right)^4 + \frac{1}{9\sqrt{3}\epsilon_n^2} \frac{9\epsilon_n^4}{2^4} \\ &= \left( \frac{1}{9 \times 2^4} + \frac{1}{3 \times 2^4} \right) \epsilon_n^2 \sqrt{3} = \frac{1}{36} \epsilon_n^2 \sqrt{3}. \end{aligned} $$

Gathering (97), (96) and (99),

$$ \int_{[021]} (x_1 - \epsilon_n) \frac{\partial c}{\partial x_1} (x_1, x_2) dx_2 dx_1 = -\frac{2}{9} \epsilon_n^2 \sqrt{3}. \quad (100) $$

The second, third and fifth equalities of (94) stem from the fact that $G$ is the barycenter of the triangle. ■

## C Proof of Corollary 43

We can now prove Corollary 43 that is useful for the computation of the generator in Section 4.5.4:

*Proof of Corollary 43.* The first equality is just the definition of $\ast d\phi$, see (49).

Let us consider the second equality. For a cycle $\sigma \in \mathfrak{C}_{1,4}$, let us consider the term

$$ \begin{aligned} \epsilon_n^{-2} \sum_{\tau \in S_2^n} \langle \partial_2 \tau, \phi \rangle^2 w(\sigma, \partial_2 \tau) &= \epsilon_n^{-2} \sum_{e \in \sigma} (\langle \partial_2 \tau^+, \phi \rangle^2 + \langle \partial_2 \tau^-, \phi \rangle^2) \\ &= 2 \epsilon_n^2 \sum_{e \in \sigma} \langle \partial_2 \tau^+, \phi \rangle^2 \end{aligned} $$

where $\tau^+$ and $\tau^-$ are the triangles adjacent to the edge $e \in \sigma$. Without loss of generality, consider $e = [01]$ in the notation of Fig. 6. We deduce from (65) that:

$$ \langle \partial_2 \tau^+, \phi \rangle^2 = \frac{3}{2} \epsilon_n^3 \int_0^{2\epsilon_n} (\phi_1^2(x_1, 0) - \phi_2^1(x_1, 0))^2 dx_1 + O(\epsilon_n^5). $$
As a consequence:

$$
\begin{align*}
\epsilon_n^{-2} \sum_{\tau \in S_2^n} \langle \partial_2 \tau, \phi \rangle^2 w(\sigma, \partial_2 \tau) &= 2\epsilon_n^{-2} \sum_{e \in \sigma} \left[ \frac{3}{2} \epsilon_n^3 \int_e (\phi_1^2 - \phi_2^1)^2 \, de + O(\epsilon_n^5) \right] \\
&= 3\epsilon_n \sum_{e \in \sigma} \int_e (\phi_1^2 - \phi_2^1)^2 \, de + O(\epsilon_n^2 \|\sigma\| \epsilon_1).
\end{align*}
$$

The first term can be interpreted as the integral of $e \mapsto \int_e (\phi_1^2 - \phi_2^1) \, de$ with respect to the measure $\epsilon_n \sum_{e \in \sigma} \delta_e$ that puts a weight $\epsilon_n$ on each edge of the cycle $\sigma$. Under Conjecture 33, we can conclude when we consider the cycle $X_u^{(n)}$ at a time $u$ instead of $\sigma$. This proves the second equality.

The last equality is a consequence of Proposition 41.

<figure><img src="image_18.png" /></figure>

References

[1] G. Akinwande and M. Reitzner. Multivariate central limit theorems for random simplicial complexes. arXiv:1912.00975, 2019.

[2] D. Aldous. Stopping times and tightness. *Annals of Probability*, 6:335-340, 1978.

[3] M.A. Armstrong. *Basic topology*. Springer, New-York, 1983.

[4] D. Bakry. Étude des transformations de Riesz dans les variétés riemanniennes à courbure de Ricci minorée. In *Séminaire de probabilités de Strasbourg*, volume 21, pages 137–172, 1987.

[5] M. Berger and B. Gostiaux. *Differential Geometry: Manifolds, Curves and Surfaces*, volume 115 of *Graduate Texts in Mathematics*. Springer, New York, 1988.

[6] P. Billingsley. *Probability and Measure*. John Wiley & Sons, New York, 3 edition, 1995.

[7] P. Billingsley. *Convergence of Probability Measures*. Wiley Series in Probability and Statistics. John Wiley & Sons, Inc., New York, second edition edition, 1999.

[8] J.-D. Boissonnat, F. Chazal, and M. Yvinec. *Geometric and Topological Inference*. Cambridge University Press, 2018.

[9] H. Brézis. *Analyse fonctionnelle, Théorie et applications*. Dunod, 2000.

[10] O. Catoni. Rough large deviation estimates for Simulated Annealing: Application to exponential schedules. *The Annals of Probability*, 20(3):1109 – 1146, 1992.

[11] R.R. Coifman and S. Lafon. Diffusion maps. *Applied and Computational Harmonic Analysis*, 21(1):5–30, 2006. Special Issue: Diffusion Maps and Wavelets.

[12] L. Decreusefond, E. Ferraz, H. Randriam, and A. Vergne. Simplicial homology of random configurations. *Adv. in Appl. Probab.*, 46:1–20, 2014.

[13] K. Devriendt and P. Van Mieghem. The simplex geometry of graphs. *Journal of Complex Networks*, 7(4):469–490, 2019.

[14] Persi Diaconis and Daniel Stroock. Geometric Bounds for Eigenvalues of Markov Chains. *The Annals of Applied Probability*, 1(1):36 – 61, 1991.

[15] B. Eckmann. Harmonische funktionen und randwertaufgaben in einem komplex. *Comm. Math. Helv.*, 17(1):240–255, 1944.

[16] K.D. Elworthy, Y. Le Jan, and X.-M. Li. *On the Geometry of Diffusion Operators and Stochastic Flows*, volume 1720 of *Lecture Notes in Mathematics (LNM)*. Springer, 1999.

[17] S.N. Ethier and T.G. Kurtz. *Markov Processus, Characterization and Convergence*. John Wiley & Sons, New York, 1986.

[18] H. Federer. *Geometric measure theory*. Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer-Verlag New York Inc., New York, 1969.

[19] R. Forman. Combinatorial differential topology and geometry. In *New perspectives in algebraic combinatorics (Berkeley, CA, 1996–97)*, volume 38 of *Math. Sci. Res. Inst. Publ.*, pages 177–206. Cambridge Univ. Press, Cambridge, 1999.

[20] Joel Friedman. Computing betti numbers via combinatorial laplacians. In *Proceedings of the Twenty-Eighth Annual ACM Symposium on Theory of Computing*, STOC '96, page 386391, New York, NY, USA, 1996. Association for Computing Machinery.

[21] E. Giné and V. Koltchinskii. *Empirical graph Laplacian approximation of Laplace-Beltrami operators: Large sample results*, volume Number 51 of *Lecture Notes-Monograph Series*, pages 238–259. Institute of Mathematical Statistics, Beachwood, Ohio, USA, 2006.
As a consequence:

$$
\begin{align*}
\epsilon_n^{-2} \sum_{\tau \in S_2^n} \langle \partial_2 \tau, \phi \rangle^2 w(\sigma, \partial_2 \tau) &= 2\epsilon_n^{-2} \sum_{e \in \sigma} \left[ \frac{3}{2} \epsilon_n^3 \int_e (\phi_1^2 - \phi_2^1)^2 de + O(\epsilon_n^5) \right] \\
&= 3\epsilon_n \sum_{e \in \sigma} \int_e (\phi_1^2 - \phi_2^1)^2 de + O(\epsilon_n^2 \|\sigma\| \epsilon_1).
\end{align*}
$$

The first term can be interpreted as the integral of $e \mapsto \int_e (\phi_1^2 - \phi_2^1) de$ with respect to the measure $\epsilon_n \sum_{e \in \sigma} \delta_e$ that puts a weight $\epsilon_n$ on each edge of the cycle $\sigma$. Under Conjecture 33, we can conclude when we consider the cycle $X_u^{(n)}$ at a time $u$ instead of $\sigma$. This proves the second equality.

The last equality is a consequence of Proposition 41.

<figure><img src="image_20.png" /></figure>

References

[1] G. Akinwande and M. Reitzner. Multivariate central limit theorems for random simplicial complexes. arXiv:1912.00975, 2019.

[2] D. Aldous. Stopping times and tightness. *Annals of Probability*, 6:335–340, 1978.

[3] M.A. Armstrong. *Basic topology*. Springer, New-York, 1983.

[4] D. Bakry. Étude des transformations de Riesz dans les variétés riemanniennes à courbure de Ricci minorée. In *Séminaire de probabilités de Strasbourg*, volume 21, pages 137–172, 1987.

[5] M. Berger and B. Gostiaux. *Differential Geometry: Manifolds, Curves and Surfaces*, volume 115 of *Graduate Texts in Mathematics*. Springer, New York, 1988.

[6] P. Billingsley. *Probability and Measure*. John Wiley & Sons, New York, 3 edition, 1995.

[7] P. Billingsley. *Convergence of Probability Measures*. Wiley Series in Probability and Statistics. John Wiley & Sons, Inc., New York, second edition edition, 1999.

[8] J.-D. Boissonnat, F. Chazal, and M. Yvinec. *Geometric and Topological Inference*. Cambridge University Press, 2018.

[9] H. Brézis. *Analyse fonctionnelle, Théorie et applications*. Dunod, 2000.

[10] O. Catoni. Rough large deviation estimates for Simulated Annealing: Application to exponential schedules. *The Annals of Probability*, 20(3):1109 – 1146, 1992.

[11] R.R. Coifman and S. Lafon. Diffusion maps. *Applied and Computational Harmonic Analysis*, 21(1):5–30, 2006. Special Issue: Diffusion Maps and Wavelets.

[12] L. Decreusefond, E. Ferraz, H. Randriam, and A. Vergne. Simplicial homology of random configurations. *Adv. in Appl. Probab.*, 46:1–20, 2014.

[13] K. Devriendt and P. Van Mieghem. The simplex geometry of graphs. *Journal of Complex Networks*, 7(4):469–490, 2019.

[14] Persi Diaconis and Daniel Stroock. Geometric Bounds for Eigenvalues of Markov Chains. *The Annals of Applied Probability*, 1(1):36 – 61, 1991.

[15] B. Eckmann. Harmonische funktionen und randwertaufgaben in einem komplex. *Comm. Math. Helv.*, 17(1):240–255, 1944.

[16] K.D. Elworthy, Y. Le Jan, and X.-M. Li. *On the Geometry of Diffusion Operators and Stochastic Flows*, volume 1720 of *Lecture Notes in Mathematics (LNM)*. Springer, 1999.

[17] S.N. Ethier and T.G. Kurtz. *Markov Processus, Characterization and Convergence*. John Wiley & Sons, New York, 1986.

[18] H. Federer. *Geometric measure theory*. Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer-Verlag New York Inc., New York, 1969.

[19] R. Forman. Combinatorial differential topology and geometry. In *New perspectives in algebraic combinatorics (Berkeley, CA, 1996–97)*, volume 38 of *Math. Sci. Res. Inst. Publ.*, pages 177–206. Cambridge Univ. Press, Cambridge, 1999.

[20] Joel Friedman. Computing betti numbers via combinatorial laplacians. In *Proceedings of the Twenty-Eighth Annual ACM Symposium on Theory of Computing*, STOC '96, page 386391, New York, NY, USA, 1996. Association for Computing Machinery.

[21] E. Giné and V. Koltchinskii. *Empirical graph Laplacian approximation of Laplace-Beltrami operators: Large sample results*, volume Number 51 of *Lecture Notes-Monograph Series*, pages 238–259. Institute of Mathematical Statistics, Beachwood, Ohio, USA, 2006.
[22] J. Grote, Z. Kabluchko, and C. Thäle. Limit theorems for random simplices in high dimensions. *ALEA, Lat. Am. J. Probab. Math. Stat.*, 16:141–177, 2019.

[23] H. Guérin, D.T. Nguyen, and V.C. Tran. Strong uniform convergence of Laplacians of random geometric and directed kNN graphs on compact manifolds. 2022. arXiv:2212.10287.

[24] A. Hatcher. *Algebraic topology*. Cambridge University Press, 2002.

[25] M. Hein, J.-Y. Audibert, and U. von Luxburg. From graphs to manifolds – weak and strong pointwise consistency of graph laplacians. In Peter Auer and Ron Meir, editors, *Learning Theory*, pages 470–485, Berlin, Heidelberg, 2005. Springer Berlin Heidelberg.

[26] N. Ikeda and S. Watanabe. *Stochastic Differential Equations and Diffusion Processes*, volume 24. North-Holland Publishing Company, 1989. Second Edition.

[27] J. Jacod and A.N. Shiryaev. *Limit Theorems for Stochastic Processes*. Springer-Verlag, Berlin, 1987.

[28] A. Jakubowski. On the Skorokhod topology. *Annales de l'Institut Henri Poincaré*, 22(3):263–285, 1986.

[29] A. Joffe and M. Métivier. Weak convergence of sequences of semimartingales with applications to multitype branching processes. *Advances in Applied Probability*, 18:20–65, 1986.

[30] M. Kahle. Topology of random clique complexes. *Discrete Mathematics*, 309:1658–1671, 2009.

[31] M. Kahle and E. Meckes. Limit theorems for betti numbers of random simplicial complexes. *Homology, Homotopy and Applications*, 15:343–374, 2013. Erratum: 18 (2016), 129–142.

[32] T. Kaufman and I. Oppenheim. High order random walks: beyond spectral gap. *Combinatorica*, 40(2):245–281, 2020.

[33] X.-D. Li. Martingale transforms and $l^p$-norm estimates of Riesz transforms on complete Riemannian manifolds. *Probab. Theory Relat. Fields*, 141:247–281, 2008.

[34] P. Malliavin. Formules de la moyenne, calcul de perturbations et théorèmes d'annulation pour les formes harmoniques. *Journal of Functional Analysis*, 17(3):274–291, 1974.

[35] K. Mehlhorn and D. Michail. Minimum cycle bases: Faster and simpler. *ACM Trans. Algorithms*, 6(1), 2010.

[36] S. Méléard and V.C. Tran. Nonlinear historical superprocess approximations for population models with past dependence. *Electronic Journal of Probability*, 17(47), 2012.

[37] S.P. Meyn and R.L. Tweedie. Stability of Markovian processes III: Foster-Lyapunov criteria for continuous-time processes. *Advances in Applied Probability*, 25(3):518–548, 1993.

[38] A. Muhammad and M. Egerstedt. Control using higher order Laplacians in network topologies. In *Proceedings of the 17th International Symposium on Mathematical Theory of Networks and Systems, Kyoto, Japan*, July 2006.

[39] S. Mukherjee and J. Steenbergen. Random walks on simplicial complexes and harmonics. *Random Structures & Algorithms*, 48(2):379–405, 2016.

[40] J.R. Munkres. *Elements of algebraic topology*. Addison-Wesley Publishing Company, Menlo Park, CA, 1984.

[41] T. Owada, G. Samorodnitsky, and G. Thoppe. Limit theorems for topological invariants of the dynamic multi-parameter simplicial complex. arXiv:2001.06860v1, 2020.

[42] L. Page, S. Brin, R. Motwani, and T. Winograd. The PageRank citation ranking: Bringing order to the Web. In *Proceedings of the 7th International World Wide Web Conference*, pages 161–172, Brisbane, Australia, 1998.

[43] O. Parzanchevski and R. Rosenthal. Simplicial complexes: spectrum, homology and random walks. *Random Structures & Algorithms*, 50(2):225–261, 2017.

[44] R. Rosenthal. Simplicial branching random walks and their applications. arXiv:1412.5406, 2014.

[45] W. Rudin. *Real and Complex Analysis*. McGraw-Hill International Editions, third edition edition, 1987.

[46] M.T. Schaub, A.R. Benson, P. Horn, G. Lippner, and A. Jadbabaie. Random walks on simplicial complexes and the normalized hodge 1-laplacian. *SIAM Review*, 62(2):353–391, 2020.

[47] A. Singer. From graph to manifold Laplacian: the convergence rate. *Applied and Computational Harmonic Analysis*, 21:128–134, 2006.

[48] D. Ting, L. Huang, and M. Jordan. An Analysis of the Convergence of Graph Laplacians. In *Proceedings of the 27th International Conference on Machine Learning*, 2010.

[49] N. García Trillos, M. Gerlach, M. Hein, and D. Slepcev. Error estimates for spectral convergence of the graph Laplacian on random geometric graphs toward the Laplace-Beltrami operator. *Foundations of Computational Mathematics*, 20:827–887, 2020.

[50] A. Trouvé. Cycle decompositions and simulated annealing. *SIAM Journal on Control and Optimization*, 34(3):966–986, 1996.

[51] J. van Neerven and R. Versendaal. $l^p$-analysis of the HodgeDirac operator associated with Witten laplacians on complete riemannian manifolds. *J Geom Anal*, 28:3109–3138, 2018.

[52] A. Vergne, I. Flint, L. Decreusefond, and P. Martins. Disaster recovery in wireless networks: a homology-based algorithm. In *Telecommunications (ICT), 2014 21st International Conference on*, pages 226–230. IEEE, 2014.

[53] U. von Luxburg, M. Belkin, and O. Bousquet. Consistency of spectral clustering. *Annals of Statistics*, 36(2):555–586, April 2008.
[22] J. Grote, Z. Kabluchko, and C. Thäle. Limit theorems for random simplices in high dimensions. *ALEA, Lat. Am. J. Probab. Math. Stat.*, 16:141–177, 2019.

[23] H. Guérin, D.T. Nguyen, and V.C. Tran. Strong uniform convergence of Laplacians of random geometric and directed kNN graphs on compact manifolds. 2022. arXiv:2212.10287.

[24] A. Hatcher. *Algebraic topology*. Cambridge University Press, 2002.

[25] M. Hein, J.-Y. Audibert, and U. von Luxburg. From graphs to manifolds – weak and strong pointwise consistency of graph laplacians. In Peter Auer and Ron Meir, editors, *Learning Theory*, pages 470–485, Berlin, Heidelberg, 2005. Springer Berlin Heidelberg.

[26] N. Ikeda and S. Watanabe. *Stochastic Differential Equations and Diffusion Processes*, volume 24. North-Holland Publishing Company, 1989. Second Edition.

[27] J. Jacod and A.N. Shiryaev. *Limit Theorems for Stochastic Processes*. Springer-Verlag, Berlin, 1987.

[28] A. Jakubowski. On the Skorokhod topology. *Annales de l'Institut Henri Poincaré*, 22(3):263–285, 1986.

[29] A. Joffe and M. Métivier. Weak convergence of sequences of semimartingales with applications to multitype branching processes. *Advances in Applied Probability*, 18:20–65, 1986.

[30] M. Kahle. Topology of random clique complexes. *Discrete Mathematics*, 309:1658–1671, 2009.

[31] M. Kahle and E. Meckes. Limit theorems for betti numbers of random simplicial complexes. *Homology, Homotopy and Applications*, 15:343–374, 2013. Erratum: 18 (2016), 129–142.

[32] T. Kaufman and I. Oppenheim. High order random walks: beyond spectral gap. *Combinatorica*, 40(2):245–281, 2020.

[33] X.-D. Li. Martingale transforms and $l^p$-norm estimates of Riesz transforms on complete Riemannian manifolds. *Probab. Theory Relat. Fields*, 141:247–281, 2008.

[34] P. Malliavin. Formules de la moyenne, calcul de perturbations et théorèmes d'annulation pour les formes harmoniques. *Journal of Functional Analysis*, 17(3):274–291, 1974.

[35] K. Mehlhorn and D. Michail. Minimum cycle bases: Faster and simpler. *ACM Trans. Algorithms*, 6(1), 2010.

[36] S. Méléard and V.C. Tran. Nonlinear historical superprocess approximations for population models with past dependence. *Electronic Journal of Probability*, 17(47), 2012.

[37] S.P. Meyn and R.L. Tweedie. Stability of Markovian processes III: Foster-Lyapunov criteria for continuous-time processes. *Advances in Applied Probability*, 25(3):518–548, 1993.

[38] A. Muhammad and M. Egerstedt. Control using higher order Laplacians in network topologies. In *Proceedings of the 17th International Symposium on Mathematical Theory of Networks and Systems, Kyoto, Japan*, July 2006.

[39] S. Mukherjee and J. Steenbergen. Random walks on simplicial complexes and harmonics. *Random Structures & Algorithms*, 48(2):379–405, 2016.

[40] J.R. Munkres. *Elements of algebraic topology*. Addison-Wesley Publishing Company, Menlo Park, CA, 1984.

[41] T. Owada, G. Samorodnitsky, and G. Thoppe. Limit theorems for topological invariants of the dynamic multi-parameter simplicial complex. arXiv:2001.06860v1, 2020.

[42] L. Page, S. Brin, R. Motwani, and T. Winograd. The PageRank citation ranking: Bringing order to the Web. In *Proceedings of the 7th International World Wide Web Conference*, pages 161–172, Brisbane, Australia, 1998.

[43] O. Parzanchevski and R. Rosenthal. Simplicial complexes: spectrum, homology and random walks. *Random Structures & Algorithms*, 50(2):225–261, 2017.

[44] R. Rosenthal. Simplicial branching random walks and their applications. arXiv:1412.5406, 2014.

[45] W. Rudin. *Real and Complex Analysis*. McGraw-Hill International Editions, third edition edition, 1987.

[46] M.T. Schaub, A.R. Benson, P. Horn, G. Lippner, and A. Jadbabaie. Random walks on simplicial complexes and the normalized hodge 1-laplacian. *SIAM Review*, 62(2):353–391, 2020.

[47] A. Singer. From graph to manifold Laplacian: the convergence rate. *Applied and Computational Harmonic Analysis*, 21:128–134, 2006.

[48] D. Ting, L. Huang, and M. Jordan. An Analysis of the Convergence of Graph Laplacians. In *Proceedings of the 27th International Conference on Machine Learning*, 2010.

[49] N. García Trillos, M. Gerlach, M. Hein, and D. Slepcev. Error estimates for spectral convergence of the graph Laplacian on random geometric graphs toward the Laplace-Beltrami operator. *Foundations of Computational Mathematics*, 20:827–887, 2020.

[50] A. Trouvé. Cycle decompositions and simulated annealing. *SIAM Journal on Control and Optimization*, 34(3):966–986, 1996.

[51] J. van Neerven and R. Versendaal. $l^p$-analysis of the HodgeDirac operator associated with Witten laplacians on complete riemannian manifolds. *J Geom Anal*, 28:3109–3138, 2018.

[52] A. Vergne, I. Flint, L. Decreusefond, and P. Martins. Disaster recovery in wireless networks: a homology-based algorithm. In *Telecommunications (ICT), 2014 21st International Conference on*, pages 226–230. IEEE, 2014.

[53] U. von Luxburg, M. Belkin, and O. Bousquet. Consistency of spectral clustering. *Annals of Statistics*, 36(2):555–586, April 2008.
[54] Ulrike von Luxburg, Agnes Radl, and Matthias Hein. Hitting and commute times in large random neighborhood graphs. *Journal of Machine Learning Research*, 15(52):1751–1798, 2014.

[55] J.B. Walsh. An introduction to stochastic partial differential equations. In *École d'été de probabilités de Saint-Flour, XIV–1984*, volume 1180 of *Lecture Notes in Math.*, pages 265–439. Springer, Berlin, 1986.

[56] E. Witten. Supersymmetry and Morse theory. *Journal of Differential Geometry*, 17(4):661–692, 1982.

[57] Dominik M. Wittmann, Daniel Schmidl, Florian Blchl, and Fabian J. Theis. Reconstruction of graphs based on random walks. *Theoretical Computer Science*, 410(38):3826–3838, 2009.

[58] Luh Yen, Denis Vanvyve, Fabien Wouters, Franois Fouss, Michel Verleysen, and Marco Saerens. Clustering using a random-walk based distance measure. In *ESANN proceedings*, pages 317–324, 01 2005.

[59] D. Yogeshwaran and R.J. Adler. On the topology of random complexes built over stationary point processes. *The Annals of Applied Probability*, 25(6):3338–3380, 2015.

[60] Z. Zhang. *Random walk on simplicial complexes*. PhD thesis, Institut polytechnique de Paris, 2020.

[61] Dengyong Zhou and Bernhard Schölkopf. Learning from labeled and unlabeled data using random walks. In Carl Edward Rasmussen, Heinrich H. Bülthoff, Bernhard Schölkopf, and Martin A. Giese, editors, *Pattern Recognition*, pages 237–244, Berlin, Heidelberg, 2004. Springer Berlin Heidelberg.

[62] V. Zobel. Spectral analysis of the Hodge Laplacian on discrete manifolds. Master's thesis, Technische Universität Berlin (J. M. Sullivan) and ZIB (I. Hotz), 2010.
[54] Ulrike von Luxburg, Agnes Radl, and Matthias Hein. Hitting and commute times in large random neighborhood graphs. *Journal of Machine Learning Research*, 15(52):1751–1798, 2014.

[55] J.B. Walsh. An introduction to stochastic partial differential equations. In *École d'été de probabilités de Saint-Flour, XIV–1984*, volume 1180 of *Lecture Notes in Math.*, pages 265–439. Springer, Berlin, 1986.

[56] E. Witten. Supersymmetry and Morse theory. *Journal of Differential Geometry*, 17(4):661–692, 1982.

[57] Dominik M. Wittmann, Daniel Schmidl, Florian Blchl, and Fabian J. Theis. Reconstruction of graphs based on random walks. *Theoretical Computer Science*, 410(38):3826–3838, 2009.

[58] Luh Yen, Denis Vanvyve, Fabien Wouters, Franois Fouss, Michel Verleysen, and Marco Saerens. Clustering using a random-walk based distance measure. In *ESANN proceedings*, pages 317–324, 01 2005.

[59] D. Yogeshwaran and R.J. Adler. On the topology of random complexes built over stationary point processes. *The Annals of Applied Probability*, 25(6):3338–3380, 2015.

[60] Z. Zhang. *Random walk on simplicial complexes*. PhD thesis, Institut polytechnique de Paris, 2020.

[61] Dengyong Zhou and Bernhard Schölkopf. Learning from labeled and unlabeled data using random walks. In Carl Edward Rasmussen, Heinrich H. Bülthoff, Bernhard Schölkopf, and Martin A. Giese, editors, *Pattern Recognition*, pages 237–244, Berlin, Heidelberg, 2004. Springer Berlin Heidelberg.

[62] V. Zobel. Spectral analysis of the Hodge Laplacian on discrete manifolds. Master's thesis, Technische Universität Berlin (J. M. Sullivan) and ZIB (I. Hotz), 2010.