# Short, Quantitative Construction of the IIC in Planar Percolation

Malo Hillairet

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Malo Hillairet. Short, Quantitative Construction of the IIC in Planar Percolation. 2025. hal-05269665v2

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Distributed under a Creative Commons Attribution 4.0 International License
# Short, Quantitative Construction of the IIC in Planar Percolation

Malo Hillairet *

## Abstract

It is standard to construct the Incipient Infinite Cluster as the limit of the critical percolation measure conditioned on 0 being connected to radius $n$, as $n$ tends to infinity. We provide a short proof of that convergence in the planar setting. A key step in the proof is to introduce an unbiased percolation configuration above which are coupled two percolations conditioned on 0 being connected to different radii. It implies a speed of convergence in total variation distance to the IIC measure upper-bounded by the dual one-arm probability, which is the first occurrence of an explicit upper-bound.

## 1 Introduction

Bernoulli percolation has been introduced in 1957 by Broadbent and Hammersley [BH57] and refers to models where, in a graph, vertices or edges are independently declared ‘open’ with some probability $p$, and the objects of interest are the macroscopic connectivity properties of the open subgraph. In general percolation models on planar graphs with Euclidean geometry such as bond percolation on $\mathbb{Z}^2$, it is well-established that there is a phase transition from parameters $p < p_c$, at which there are no macroscopic open connected component, to parameters $p > p_c$, at which there is a connected component of positive density. At the critical parameter $p_c \in (0, 1)$, there is no infinite connected component, but there are macroscopic connected components with a rich geometry. In this short article, we provide a new proof of existence of the so-called Incipient Infinite Cluster (IIC) in critical planar Bernoulli percolation models. Although this object had been introduced before, it is Kesten [Kes86] who rigorously established its existence on graphs similar to $\mathbb{Z}^2$ as the limit of the sequence of measures

$$ \lim_{n \to +\infty} \mathbf{P}_{p_c} (\cdot | 0 \text{ is connected to distance } n) \qquad (1) $$

where $\mathbf{P}_p$ denotes Bernoulli percolation at parameter $p$ and $p_c$ is the critical parameter of the model, and also as

$$ \lim_{\substack{p \to p_c \\ p > p_c}} \mathbf{P}_p (\cdot | 0 \leftrightarrow \infty), $$

where $0 \leftrightarrow \infty$ means there is an infinite connected component containing 0. The IIC thus corresponds to the percolation measure conditioned on 0 being in an infinite connected component at $p_c$, which is a relevant object in view of understanding better the macroscopic connected components at criticality. There are other equivalent definitions of the IIC, for instance in the works of Járai [Jár03] and Borgs, Chayes, Kesten, Spencer [Bor+01] where the IIC is constructed by considering the largest cluster in the box of size $n$ around the origin and translating it so that it contains 0. It was also shown by Hammond, Pete and Schramm [HPS15] that dynamical percolation observed at a typical exceptional time has the distribution of the IIC (dynamical percolation is the continuous-time process where the states of sites in critical percolation are resampled at exponential rates, and exceptional times are times at which 0 is connected to infinity by an open path). For background on dynamical percolation and exceptional times, we refer to a survey by Steif [Ste09]. A natural question left is the speed of convergence in (1), this is the main topic of this article.

**Previously known speeds of convergence.** Consider the model of face percolation on the hexagonal lattice, that is, consisting in regular hexagons paving the plane. We independently color the hexagons in black with probability $\frac{1}{2}$ and in white otherwise (as $\frac{1}{2}$ is the critical parameter in this model by Kesten's theorem [Kes80]). We denote by $\mathbf{P} = \mathbf{P}_{1/2}$ the associated probability distribution.

<footnote>*Institut Fourier, Université Grenoble Alpes.</footnote>
For $n \in \mathbb{N}$ we write $\Lambda_n$ for the set of hexagons at distance at most $n$ to 0, and $\partial\Lambda_n$ for its outer boundary $\Lambda_{n+1} \setminus \Lambda_n$. We denote by $\Omega_{\Lambda_n}$ the set $\{0, 1\}^{\Lambda_n}$ of percolation configurations on $\Lambda_n$. We let $\{0 \leftrightarrow \partial\Lambda_n\} \subseteq \Omega_{\Lambda_n}$ be the event that there is a path of black hexagons, in $\Lambda_n$, starting at a neighbor of 0 and ending at an hexagon adjacent to $\partial\Lambda_n$. We call $\{0 \leftrightarrow \partial\Lambda_n\}$ the **one-arm event** to radius $n$. Kesten's initial method for constructing the IIC is called the transfer matrix method. Although not explicitly stated in [Kes86], it implies the following speed of convergence in total variation distance, of the sequence of measures $\mathbf{P}(\cdot|0 \leftrightarrow \partial\Lambda_n)$, $n \in \mathbb{N}$.

**Theorem 1.1** ([Kes86], Transfer-matrix-method speed of convergence). For all $\eta \in (0, \frac{1}{2})$, for all $k \le m \le n \in \mathbb{N}$ such that $\frac{m}{k}$ is large enough,

$$ \sup_{E \subseteq \Omega_{\Lambda_k}} \left| \mathbf{P}(E|0 \leftrightarrow \partial \Lambda_m) - \mathbf{P}(E|0 \leftrightarrow \partial \Lambda_n) \right| \le \exp\left(-\log(m/k)^{\frac{1}{2}-\eta}\right), $$

where the supremum is taken over events E depending only on hexagons in $\Lambda_k$.

With the same proof, which only relies on ‘soft’ tools of planar percolation such as RSW estimates and the Harris-FKG inequality, this above result extends to similar planar Bernoulli percolation on planar transitive graphs, notably bond percolation on $\mathbb{Z}^2$. The transfer matrix method has been used and extended to construct multiple-arms IIC by Damron and Sapozhnikov [DS11], IIC in a larger class of graphs such as slabs of $\mathbb{Z}^2$ by Basu and Sapozhnikov [BS17] and recently IIC in higher dimensions by Chatterjee, Chinmay, Hanson and Sosoe [Cha+25]. In an unpublished work by Schramm [Sch] in the case of the 1-arm and in [GPS13, Section 3] in the case of the 4-arm, Schramm and Garban, Pete and Schramm developed a proof of the convergence in (1), not using transfer matrices and yielding a better speed of convergence.

**Theorem 1.2** ([Sch; GPS13], Scale-by-scale coupling speed of convergence). There exists a positive constant $c$ satisfying that for all $k \le m \le n \in \mathbb{N}$ such that $\frac{m}{k}$ is large enough,

$$ \sup_{E \subseteq \Omega_{\Lambda_k}} \left| \mathbf{P}(E|0 \leftrightarrow \partial \Lambda_m) - \mathbf{P}(E|0 \leftrightarrow \partial \Lambda_n) \right| \le \left(\frac{k}{m}\right)^c. $$

The constant $c$ in Theorem 1.2 can be thought of as an RSW constant, and it would be tedious to rely on the construction in [GPS13] to obtain an explicit lower-bound on $c$. The following question is left open, where ‘largest’ can be understood in the sense of a supremum.

**Question 1.** What is the largest $c$ satisfying the conclusion of Theorem 1.2, and can it be expressed in terms of arm exponents?

We briefly expand on the difference with the construction of [Kes86] to highlight the progression from the proof of Theorem 1.1 in [Kes86] to the proof of Theorem 1.2 in [Sch; GPS13], consisting in an ‘annulus-by-annulus’ exploration, to the method of the present article where we use a ‘site-by-site’ exploration. The transfer matrix method of [Kes86] relies on considering annuli of width $\exp(i^\alpha)$, with $\alpha > 1$, and under the assumption that one can find a black circuit surrounding $\Lambda_k$ in all of the annuli between $\Lambda_k$ and $\Lambda_m$. The proof of [Sch; GPS13] uses two additional elements. The first is removing the condition that a black circuit needs to be found in all of the annuli. Instead, we consider annuli of width $\exp(i)$ and it is shown there is a constant probability of achieving coupling at each of these annuli regardless of what has been revealed in the previous annuli. The second element is to try achieving coupling by revealing an ‘innermost black circuit’ in each of the annuli, which is more explicit than going through a transfer matrix. The core of the present article is to find another way of coupling, which allows us to reveal an outermost black circuit in $\Lambda_m$, thus considering $\Lambda_m \setminus \Lambda_k$ as a very large annulus. As a consequence, we get rid of the need to split that region into nested annuli and obtain a more quantitative speed of convergence.

**Main result.** The main contribution of this article is to provide a short and simple proof of existence of the IIC using standard arguments of percolation theory and statistical mechanics. This partially answers Question 1 by deriving a lower-bound on $c$ equal to the one-arm exponent. This is the first time a lower-bound on $c$ is made explicit. To this end, we provide a coupling construction relying on a site-by-site exploration. We explain the proof strategy in more detail soon after stating the result.

**Theorem 1.3** (Site-by-site coupling speed of convergence). For all $k \le m \le n \in \mathbb{N}^*$,

$$ \sup_{E \subseteq \Omega_{\Lambda_k}} \left| \mathbf{P}(E|0 \leftrightarrow \partial \Lambda_m) - \mathbf{P}(E|0 \leftrightarrow \partial \Lambda_n) \right| \le \mathbf{P}(\Lambda_k \leftrightarrow^* \partial \Lambda_m), $$
where {$\Lambda_k \leftrightarrow^* \Lambda_m$} is the dual one-arm event that there is a path of white hexagons from $\Lambda_k$ to $\partial\Lambda_m$. In particular, for all $\eta \in (0, \frac{5}{48})$, for all $k \le m \le n \in \mathbb{N}$ such that $\frac{m}{k}$ is large enough,

$$ \sup_{E \subseteq \Omega_{\Lambda_k}} \left| \mathbf{P}(E|0 \leftrightarrow \partial \Lambda_m) - \mathbf{P}(E|0 \leftrightarrow \partial \Lambda_n) \right| \le \left(\frac{k}{m}\right)^{\frac{5}{48}-\eta} $$

The first part of the theorem, not mentioning the one-arm exponent, holds with the same proof in any planar model, regardless of self-duality or symmetry. The value $\frac{5}{48}$ appearing in the second part of the Theorem is called the one-arm exponent. Indeed, Lawler, Schramm and Werner [LSW02] proved $\mathbf{P}(\Lambda_k \leftrightarrow \partial \Lambda_m) = (k/m)^{5/48+o(1)}$ when $\frac{m}{k}$ goes to infinity. The same result is conjectured, but not proved, in critical bond percolation on $\mathbb{Z}^2$.

**Proof overview.** The probabilistic tools used in the proof are a coupling based on uniform random variables and a property reminiscent of a spatial Markov property, which already appeared in the works of Kesten and Garban, Pete and Schramm. We immediately give a sketch of proof, which could already be convincing as a full proof. We will couple $\omega^{(m)}$ a percolation conditioned on having a one-arm to distance $m$, and $\omega^{(n)}$ conditioned on having a one-arm to distance $n$. A key ingredient of the proof is to sample both $\omega^{(m)}$ and $\omega^{(n)}$ above the same unconditioned percolation $\omega$, using uniform random variables attached to the hexagons of the lattice. The role of $\omega$ is crucial to define the order in which the uniform random variables are revealed.

In that sketch of proof and in the rest of the paper, given $n \ge 1$ we denote by $\varepsilon$ a generic element of $\Omega_{\Lambda_n}$: for example, for $x \in \Lambda_n$ we abbreviate the set $\{\varepsilon \in \Omega_{\Lambda_n} : \varepsilon_x = 1\}$ by $\{\varepsilon_x = 1\}$, and will use the letter $\varepsilon$ specifically for that notation. The coupled configurations $\omega, \omega^{(m)}, \omega^{(n)}$ are related to a distinct, abstract probability space which we define later, and we will call $\mathbb{P}$ the associated probability. When using $\mathbf{P}$, the configurations $\omega, \omega^{(m)}, \omega^{(n)}$ are treated as deterministic.

*Sketch of proof of Theorem 1.3.* Let $k \le m \le n$ be fixed integers. We construct a coupling between a percolation configuration $\omega$ having law $\mathbf{P}$, $\omega^{(m)}$ having law $\mathbf{P}(\cdot|0 \leftrightarrow \partial \Lambda_m)$ and $\omega^{(n)}$ having law $\mathbf{P}(\cdot|0 \leftrightarrow \partial \Lambda_n)$, through exploring $\omega$ and assigning values to $(\omega_x, \omega_x^{(m)}, \omega_x^{(n)})_{x \in \Lambda_m}$ site by site using uniform random variables $(U_x)_{x \in \Lambda_m}$. Assume that at some point, the hexagons $x_1, x_2, \dots, x_k$ have been explored, meaning that for each $x \in \{x_1, \dots, x_k\}$, the values of $U_x, \omega_x, \omega_x^{(m)}, \omega_x^{(n)}$ are known. From that information, we pick $x_{k+1}$ in $\Lambda_m \setminus \{x_1, \dots, x_k\}$, and set the values of $\omega_{x_{k+1}}, \omega_{x_{k+1}}^{(m)}$ and $\omega_{x_{k+1}}^{(n)}$ according to

$$ \begin{aligned} \omega_{x_{k+1}} &:= 1 \{ U_{x_{k+1}} \le \frac{1}{2} \}, \\ \omega_{x_{k+1}}^{(m)} &:= 1 \{ U_{x_{k+1}} \le \mathbf{P}(\varepsilon_{x_{k+1}} = 1 | \varepsilon_{x_j} = \omega_{x_j}^{(m)} \forall j \le k, 0 \leftrightarrow \partial \Lambda_m) \}, \\ \omega_{x_{k+1}}^{(n)} &:= 1 \{ U_{x_{k+1}} \le \mathbf{P}(\varepsilon_{x_{k+1}} = 1 | \varepsilon_{x_j} = \omega_{x_j}^{(n)} \forall j \le k, 0 \leftrightarrow \partial \Lambda_n) \}. \end{aligned} $$

The criterion for choosing $x_{k+1}$ is to reveal the outermost black circuit of $\omega$ in $\Lambda_m$ first. Such a circuit exists and separates $\Lambda_k$ from $\partial\Lambda_m$ if and only if there is no white arm from $\Lambda_k$ to $\partial\Lambda_m$ in $\omega$, hence with probability $1 - \mathbf{P}(\Lambda_k \leftrightarrow^* \partial\Lambda_m)$. Using the FKG inequality, we can prove that $\omega^{(m)} \ge \omega$ and $\omega^{(n)} \ge \omega$ almost surely, implying that any black circuit in $\omega$ is also black in $\omega^{(m)}$ and $\omega^{(n)}$. We can see here the importance of $\omega$: we used it to construct an exploration revealing a black circuit common to $\omega^{(m)}$ and $\omega^{(n)}$ while making sure no hexagon has been explored inside that circuit. The final observation is that if such a black circuit $\Gamma$ cutting $\Lambda_k$ off of $\partial\Lambda_m$ has been revealed, then in the region interior to $\Gamma$, conditioning on $\{0 \leftrightarrow \partial\Lambda_m\}$ or on $\{0 \leftrightarrow \partial\Lambda_n\}$ is equivalent to conditioning on $\{0 \leftrightarrow \Gamma\}$. As a consequence and due to their definition, the variables $\omega^{(m)}$ and $\omega^{(n)}$ must coincide in the inner region defined by $\Gamma$, hence in $\Lambda_k$. $\square$

**Perspectives.** There is no reason for the dual one-arm probability to be a sharp upper-bound for the speed of convergence. In the proof, we are likely to discover more black vertices in $\omega^{(m)}$ and $\omega^{(n)}$ than just the vertices in $\omega$, so maybe the current exploration reveals a black circuit common to $\omega^{(m)}$ and $\omega^{(n)}$ before reaching $\Lambda_k$ with even larger probability. The choice of the exploration is also crucial, and there could be more clever explorations. Thus, Question 1 is still open and we expect the answer to be larger than the one-arm exponent $\frac{5}{48}$.

Another perspective is the generalization of the above proof to more arms, as the proofs of [Kes86] and [GPS13] can be adapted to arms of different colors. This however requires higher-level arguments
such as arm separation and quasi-multiplicativity. Even more importantly, when trying the site-by-site exploration proof with arms of different colors, there is no clear monotonicity with respect to the unbiased percolation configuration $\omega$. As a consequence, it is no longer easy to find a circuit with respect to which a similar spatial Markov property holds. It seems the proof of Theorem 1.3 could be adapted as such to the monochromatic two-arm event. However, for the monochromatic k-arm with $k \ge 3$, it is unclear what sets could play the role of the black circuits.

**Question 2.** Can multiple-arms IIC be constructed using a site-by-site exploration, as in the proof of Theorem 1.3?

Finally, in larger dimensions there is very low probability of finding a black hypersurface surrounding 0 at criticality. Thus, adapting the construction of Theorem 1.3 to larger dimensions looks challenging even for the one-arm.

**Question 3.** For $d \ge 3$, can the IIC of critical percolation on $\mathbb{Z}^d$ be constructed using a site-by-site exploration?

**Acknowledgements.** The author thanks Hugo Vanneuville and Vincent Beffara for instructive discussions and detailed comments on the manuscript, and Daniel de la Riva and Christophe Garban for private communications and remarks at an early stage of the project. We thank Christophe Garban for communicating the unpublished notes by Schramm [Sch].

## 2 Proofs

**Setting.** We work with site percolation on the triangular lattice $\mathbb{T}$ having $\mathbb{Z} + (\frac{1}{2}, \frac{\sqrt{3}}{2})\mathbb{Z} \subseteq \mathbb{R}^2$ as a set of vertices and where edges are between vertices at Euclidean distance 1 to each other. We represent this model by face percolation on the hexagonal lattice by centering a hexagon of diameter $\sqrt{3}$ on top of each vertex, and coloring in black the hexagons covering open sites and in white the others. We identify sets of vertices $\Lambda \subseteq \mathbb{Z} + (\frac{1}{2}, \frac{\sqrt{3}}{2})\mathbb{Z}$ with subgraphs of $\mathbb{T}$ having full set of edges. We define $\partial\Lambda$, the boundary of $\Lambda$, as the set of vertices at graph distance exactly 1 to $\Lambda$. For $n \in \mathbb{N}$, we let $\Lambda_n$ be the set of vertices at graph distance to 0 less than or equal to $n$ in $\mathbb{T}$. Given $\Lambda \subseteq \mathbb{T}$, we let $\Omega_{\Lambda} = \{0, 1\}^{\Lambda}$ be the space of percolation configurations on $\Lambda$. We endow $\Omega_{\mathbb{T}}$ with the Bernoulli percolation measure $\mathbf{P} = \text{Ber}(\frac{1}{2})^{\otimes \mathbb{T}}$. We also introduce an abstract probability space with associated probability $\mathbb{P}$, on which is defined a countable family of independent uniform random variables on $[0, 1]$. The letter $\omega$, with or without superscript, will always be used for percolation configurations defined using these uniform random variables. We reserve the letter $\eta$ to fixed elements of $\Omega_{\Lambda}$, for $\Lambda \subseteq \mathbb{T}$. If $S \subseteq \Lambda$ and $\eta \in \Omega_{\Lambda}$, we denote by $\eta_S$ the restriction of $\eta$ to $S$, and we use similar notation for random percolation configurations. We let $\varepsilon$ denote a generic element of $\Omega_{\Lambda}$: for $S \subseteq \Lambda$ and $\eta \in \Omega_S$, we let

$$ \{\varepsilon_S = \eta\} := \{\varepsilon \in \Omega_{\Lambda} : \varepsilon_S = \eta\}, \quad (2) $$

and we use the letter $\varepsilon$ specifically for that notation. Given $S, S' \subseteq \mathbb{T}$, we write $\{S \leftrightarrow S'\}$ (resp. $\{S \leftrightarrow^* S'\}$) for the event that there is a neighbor-to-neighbor path of vertices $(x_1, \dots, x_k)$ such that $x_1 \in \partial S, x_k \in \partial S'$ and $\omega_{x_i} = 1$ (resp. $\omega_{x_i} = 0$) for all $1 \le i \le k$. By convention, we declare certain the event $\{S \leftrightarrow S'\}$ if $S$ and $S'$ have non-empty intersection or are adjacent. We will also consider connection events where the connection path is constrained to stay within a region $\Lambda$, which we will denote by $\{S \stackrel{\Lambda}{\leftrightarrow} S'\}$. Finally, given $n \in \mathbb{N}$, $S \subseteq \Lambda_n$ and $\eta \in \Omega_S$, we say $\eta$ is **compatible** with $\{0 \leftrightarrow \partial\Lambda_n\}$ if there exists $\eta' \in \Omega_{\Lambda_n}$ such that $\eta'_S = \eta$ and $\eta' \in \{0 \leftrightarrow \partial\Lambda_n\}$.

**Site-by-site coupling.** We hereby describe what we call an exploration and how we sample percolation configurations conditioned on one-arm events using uniform random variables. Let $\Lambda \subseteq \mathbb{T}$ be a finite set of vertices and $(U_x)_{x \in \Lambda}$ a collection of iid $[0, 1]$-valued uniform random variables. We will denote by $X = (X(i))_{1 \le i \le |\Lambda|}$ a random bijection from $\{1, \dots, |\Lambda|\}$ to $\Lambda$, satisfying that for all $1 \le i \le |\Lambda| - 1$,

$X(i)$ is a deterministic function of $(X(j))_{1 \le j \le i-1}$ and $(U_{X(j)})_{1 \le j \le i-1}$.

We say that $X$ is an **exploration** of $\Lambda$ associated to $(U_x)_{x \in \Lambda}$. We also let $X_{[i]} := \{X(1), \dots, X(i)\}$ for $0 \le i \le |\Lambda|$, this is the set of vertices which have been explored at iteration $i$. Let $n \in \mathbb{N}$ and let $X$ be an exploration of $\Lambda$ associated to $(U_x)_{x \in \Lambda}$. We let $\omega^{(n)}$ be defined by

$$ \omega_{X(i)}^{(n)} := 1 \{ U_{X(i)} \le \mathbf{P}(\varepsilon_{X(i)} = 1 | \varepsilon_{X_{[i-1]}} = \omega_{X_{[i-1]}}^{(n)}, 0 \leftrightarrow \partial \Lambda_n ) \}, \quad (3) $$
where we recall the letter $\epsilon$ is used for denoting events as in (2) and $\omega_{X_{[i-1]}}, \omega_{X_{[i-1]}}^{(m)}, \omega_{X_{[i-1]}}^{(n)}$ and $X(i)$ are treated as deterministic when in input of the probability $\mathbf{P}$. In parallel, we define $\omega$ by

$$ \omega_{X(i)} = 1\left\{U_{X(i)} \le \frac{1}{2}\right\}. $$

This is equivalent to setting $\omega_x = 1\{U_x \le \frac{1}{2}\}$ for all $x \in \Lambda$. Hence, $\omega$ has distribution $\mathbf{P}$ and is coupled to $\omega^{(n)}$. The following lemma expresses that defining $\omega^{(n)}$ as in (3) samples it from the distribution $\mathbf{P}(\cdot|0 \leftrightarrow \partial\Lambda_n)$, and a simple use of the FKG inequality is sufficient to additionally obtain the monotonicity $\omega^{(n)} \ge \omega$. This lemma and its proof are very similar to [DRT19, Lemma 2.1].

**Lemma 2.1.** Let $\Lambda$ be a finite subset of $\mathbb{T}$. Let $n \in \mathbb{N}$ and $(U_x)_{x \in \Lambda}$ be independent $[0, 1]$-valued uniform random variables. Let $X$ be an exploration of $\Lambda$ associated to $(U_x)_{x \in \Lambda}$. Then, the random variable $\omega^{(n)} \in \Omega_{\Lambda}$ defined by (3) has as a distribution the restriction to $\Lambda$ of $\mathbf{P}(\cdot|0 \leftrightarrow \partial\Lambda_n)$. Moreover, $\omega^{(n)} \ge \omega$ almost surely.

*Proof of Lemma 2.1.* Let $n \in \mathbb{N}^*$, $\Lambda \subseteq \mathbb{T}$, $(U_x)_{x \in \Lambda}$, $(X(i))_{1 \le i \le |\Lambda|}$ be as in the statement of the lemma, and let $\omega_{\Lambda}, \omega_{\Lambda}^{(n)}$ be defined as in (3). We prove by induction on $0 \le i \le |\Lambda|$: for all $\eta \in \Omega_{\Lambda}$

$$ \mathbb{P}\left(\omega_{X[i]}^{(n)} = \eta_{X[i]}\right) = \mathbf{P}\left(\varepsilon_{X[i]} = \eta_{X[i]} \mid 0 \leftrightarrow \partial\Lambda_n\right). $$

Both probabilities are 1 at $i=0$. For $i \ge 1$, assume the above holds at $i-1$. By (3) and on the event $\{\omega_{X_{[i-1]}}^{(n)} = \eta_{X_{[i-1]}}^{(n)}\}$,

$$ \{\omega_{X(i)}^{(n)} = 1\} = \{U_{X(i)} \le \mathbf{P}(\varepsilon_{X(i)} = 1 | \varepsilon_{X_{[i-1]}} = \eta_{X_{[i-1]}}), 0 \leftrightarrow \partial\Lambda_n\}. $$

Remark that $\omega_{X_{[i-1]}}^{(n)}$ is measurable in $(X(j))_{1 \le j \le i-1}$ and the attached uniform random variables, and $X(i)$ is deterministic in this information. Therefore, conditionally on $\{\omega_{X_{[i-1]}}^{(n)} = \eta_{X_{[i-1]}}^{(n)}\}$, $U_{X(i)}$ has a uniform distribution. We deduce

$$ \mathbb{P}\left(\omega_{X(i)}^{(n)} = \eta_{X(i)} | \omega_{X_{[i-1]}}^{(n)} = \eta_{X_{[i-1]}}\right) = \mathbf{P}\left(\varepsilon_{X(i)} = \eta_{X(i)} | \varepsilon_{X_{[i-1]}} = \eta_{X_{[i-1]}}, 0 \leftrightarrow \partial\Lambda_n\right). $$

We conclude using the above and the induction hypothesis:

$$ \begin{aligned} \mathbb{P}\left(\omega_{X[i]}^{(n)} = \eta_{X[i]}\right) &= \mathbb{P}\left(\omega_{X_{[i-1]}^{(n)}} = \eta_{X_{[i-1]}}\right) \mathbb{P}\left(\omega_{X(i)}^{(n)} = \eta_{X(i)} \mid \omega_{X_{[i-1]}^{(n)}} = \eta_{X_{[i-1]}}\right) \\ &= \mathbf{P}\left(\varepsilon_{X_{[i-1]}} = \eta_{X_{[i-1]}} \mid 0 \leftrightarrow \partial\Lambda_n\right) \\ & \quad \times \mathbf{P}\left(\varepsilon_{X(i)} = \eta_{X(i)} \mid \varepsilon_{X_{[i-1]}} = \eta_{X_{[i-1]}}, 0 \leftrightarrow \partial\Lambda_n\right) \\ &= \mathbf{P}\left(\varepsilon_{X[i]} = \eta_{X[i]} \mid 0 \leftrightarrow \partial\Lambda_n\right). \end{aligned} $$

It remains to prove $\omega^{(n)} \ge \omega$. Let $x_1, \dots, x_i \in \Lambda$ be distinct and $\eta \in \Omega_{\Lambda}$. We denote by $x_{[i-1]}$ the set $\{x_1, \dots, x_{i-1}\}$. By the FKG inequality (see e.g. [Gri99, Theorem 2.4]), the measure $\mathbf{P}(\cdot|\varepsilon_{x_{[i-1]}} = \eta_{x_{[i-1]}})$ being a product measure and $\{\varepsilon_{x_i} = 1\}, \{0 \leftrightarrow \partial\Lambda_n\}$ increasing events,

$$ \frac{1}{2} \le \mathbf{P}\left(\varepsilon_{x_i} = 1 \mid \varepsilon_{x_{[i-1]}} = \eta_{x_{[i-1]}}, 0 \leftrightarrow \partial\Lambda_n\right). $$

We deduce that, almost surely,

$$ 1\left\{U_{X(i)} \le \frac{1}{2}\right\} \le 1\left\{U_{X(i)} \le \mathbf{P}\left(\varepsilon_{X(i)} = 1 \mid \varepsilon_{X_{[i-1]}} = \omega_{X_{[i-1]}}^{(n)}, 0 \leftrightarrow \partial\Lambda_n\right)\right\}, $$

and this completes the proof. $\square$
**Spatial Markov property with respect to black circuits.** An essential tool for constructing the IIC is a property of independence of the inside and the outside of a revealed black circuit of measures conditioned on having a one-arm. Indeed, if we have already revealed a black circuit cutting 0 off of $\partial\Lambda_n$, conditioning on having a one-arm from 0 to radius $n$ is exactly conditioning on having a one-arm from 0 to the circuit and a one-arm from the circuit to radius $n$, which are events depending on disjoint sets of hexagons. This observation leads to the conditional independence expressed by (16) in [Kes86], or in the upcoming lemma.

Before stating the lemma, we introduce the notion of a **circuit**: it is a finite path $\Gamma$ having adjacent endpoints and such that $T\setminus\Gamma$ has exactly two connected components. We call interior of $\Gamma$ and denote by $\text{int}(\Gamma)$ its finite connected component, and we call exterior of $\Gamma$, or $\text{ext}(\Gamma)$, the infinite one. We say that a circuit $\Gamma$ is **circling** around a set $\Lambda \subseteq T$ if $\Lambda \subseteq \text{int}(\Gamma)$. It would perhaps be more standard to define circuits as closed loops, or to use cutsets instead. However, the above definition of circuits has the advantage of making it easy to define an interior and an exterior and we avoid dealing with the possibility that $T\setminus\Gamma$ has more than two connected components.

**Lemma 2.2.** Let $n \ge 0$ and let $\Gamma$ be a circuit circling around 0 in $\Lambda_n$. Let $S$ be a finite set of vertices containing $\Gamma$ and let $\eta \in \Omega_S$ be such that $\Gamma$ is black in $\eta$ and $\eta$ is compatible with $\{0 \leftrightarrow \partial\Lambda_n\}$. Then, for any event $A$ depending only on coordinates inside $\text{int}(\Gamma)$,

$$ \mathbf{P}(A | \varepsilon_S = \eta, 0 \leftrightarrow \partial\Lambda_n) = \mathbf{P}(A | \varepsilon_S = \eta, 0 \leftrightarrow \Gamma) = \mathbf{P}(A | \varepsilon_{S \cap \text{int}(\Gamma)} = \eta_{S \cap \text{int}(\Gamma)}, 0 \leftrightarrow \Gamma). $$

*Proof of Lemma 2.2.* Let $n \in \mathbb{N}$, $\Gamma \subseteq S \subseteq \Lambda_n$, $\eta \in \Omega_S$ be as in the statement of the lemma. First observe

$$ \{\varepsilon_S = \eta, 0 \leftrightarrow \partial\Lambda_n\} = \{\varepsilon_S = \eta, 0 \stackrel{\text{int}(\Gamma)}{\leftrightarrow} \Gamma, \Gamma \stackrel{\text{ext}(\Gamma)}{\leftrightarrow} \partial\Lambda_n\}. $$

Indeed, since $\Gamma$ is black in $\eta$ and circling around 0, any black one-arm from 0 to $\partial\Lambda_n$ connects 0 to $\Gamma$ and $\Gamma$ to $\partial\Lambda_n$. Conversely, if 0 is connected to $\Gamma$ by a black path $\pi_{\text{int}}$ and $\Gamma$ is connected to $\partial\Lambda_n$ by a black path $\pi_{\text{ext}}$, the union of sets $\pi_{\text{int}} \cup \Gamma \cup \pi_{\text{ext}}$ is black and connects 0 to $\partial\Lambda_n$. Now, let $A$ be an event such that $A$ depends only on $\text{int}(\Gamma)$. By independence of percolation events related to disjoint sets,

$$ \mathbf{P}(A \cap \{0 \stackrel{\text{int}(\Gamma)}{\leftrightarrow} \Gamma\} \cap \{\Gamma \stackrel{\text{ext}(\Gamma)}{\leftrightarrow} \partial\Lambda_n\} | \varepsilon_S = \eta) = \mathbf{P}(A \cap \{0 \stackrel{\text{int}(\Gamma)}{\leftrightarrow} \Gamma\} | \varepsilon_S = \eta) \mathbf{P}(0 \stackrel{\text{ext}(\Gamma)}{\leftrightarrow} \partial\Lambda_n | \varepsilon_S = \eta). $$

By the definition of conditional expectation and applying the above to the certain event $\Omega$ as well, we deduce

$$ \mathbf{P}(A | \varepsilon_S = \eta, 0 \leftrightarrow \partial\Lambda_n) = \mathbf{P}(A | \varepsilon_S = \eta, 0 \leftrightarrow \Gamma). $$

Finally, since the events $A$ and $\{0 \leftrightarrow \Gamma\}$ depend only on $\varepsilon_{\text{int}(\Gamma)}$, we have

$$ \mathbf{P}(A \cap \{0 \leftrightarrow \Gamma\} | \varepsilon_S = \eta_S) = \mathbf{P}(A \cap \{0 \leftrightarrow \Gamma\} | \varepsilon_{S \cap \text{int}(\Gamma)} = \eta_{S \cap \text{int}(\Gamma)}), $$

which amounts to the second equality of the lemma after dividing by

$$ \mathbf{P}(0 \leftrightarrow \Gamma | \varepsilon_S = \eta_S) = \mathbf{P}(0 \leftrightarrow \Gamma | \varepsilon_{S \cap \text{int}(\Gamma)} = \eta_{S \cap \text{int}(\Gamma)}). \quad \square $$

We finally provide a proof of Theorem 1.3. It is a formalized version of the sketch of proof given in the introduction, making use of Lemmas 2.1 and 2.2.

*Proof of Theorem 1.3.* Let $k \le m \le n$ and let $(U_x)_{x \in \Lambda_m}$ be iid uniform random variables on $[0, 1]$. From these variables we construct $\omega$, $\omega^{(m)}$ and $\omega^{(n)}$: three percolation configurations in $\Lambda_m$ satisfying that $\omega$ has law $\mathbf{P}$, $\omega^{(m)}$ has law $\mathbf{P}(\cdot|0 \leftrightarrow \partial\Lambda_m)$, $\omega^{(n)}$ has law $\mathbf{P}(\cdot|0 \leftrightarrow \partial\Lambda_n)$ and $\omega^{(m)} \ge \omega$ and $\omega^{(n)} \ge \omega$. According to Lemma 2.1, it is sufficient for these properties to hold to introduce an exploration $\{X(i), 1 \le i \le |\Lambda_m|\}$ of the vertices in $\Lambda_m$, associated to $(U_x)_{x \in \Lambda_m}$, then define $\omega$, $\omega^{(m)}$, $\omega^{(n)}$ according to (3). Additionally, we define $\{X(i), 1 \le i \le |\Lambda_m|\}$ carefully so that it reveals the outermost black circuit of $\omega$ before revealing anything in its interior, as follows. For $1 \le i \le |\Lambda_m|$, let $E_{i-1}$ be the set of sites of $\Lambda_m \setminus X_{[i-1]}$ connected by a white path to $\partial\Lambda_m$ in $\omega_{X_{[i-1]}}$. We choose the next site to reveal $X(i)$ by picking sites in $E_{i-1}$ in priority: if $E_{i-1}$ is non-empty, pick $X(i)$ arbitrarily in $E_{i-1}$, otherwise pick $X(i)$ arbitrarily in the set of unrevealed sites $\Lambda_m \setminus X_{[i-1]}$ (here ‘arbitrarily in $E_{i-1}$’ means any choice of $X(i) \in E_{i-1}$ that is deterministic in $X_{[i-1]}$ and $(U_{X(j)})_{j \le i-1}$ is satisfactory). Note that $E_{i-1}$ also depends only on $X_{[i-1]}$ and $(U_{X(j)})_{j \le i-1}$ so that this is well-defined and $X$ then satisfies the conditions of Lemma 2.1.
Our goal is to prove that $\omega^{(m)}$ and $\omega^{(n)}$ agree in $\Lambda_k$ as soon as $\omega \notin \{\Lambda_k \leftrightarrow^* \partial\Lambda_m\}$. Define the stopping-time $\tau$ by

$$ \tau := \min \{0 \le i \le |\Lambda_m| : E_i = \emptyset\}. $$

Hence, $\tau$ is the first time in the exploration at which no unrevealed site is connected to $\partial\Lambda_m$ by a white path in $\omega_{X_{[\tau]}}$. From now on, assume the event $\{X_{[\tau]} \cap \Lambda_k = \emptyset\}$. This event is exactly the event that the white connected component of $\partial\Lambda_m$ in $\omega$ and its black boundary have been revealed before the exploration has reached $\Lambda_k$, so that

$$ \{X_{[\tau]} \cap \Lambda_k \neq \emptyset\} = \{\Lambda_k \leftrightarrow^* \partial\Lambda_m \text{ in } \omega\}. \quad (4) $$

Let $C$ be the connected component of 0 in $\Lambda_m \setminus X_{[\tau]}$. Note $\Lambda_k \subseteq C$ since $\{X_{[\tau]} \cap \Lambda_k = \emptyset\}$. Every site $x \in \partial C$ is in $X_{[\tau]}$ by maximality of $C$ and black in $\omega_{X_{[\tau]}}$ by minimality of $\tau$. By convention, a site neighbor to $\partial\Lambda_m$ is always said to be connected by a white path to $\partial\Lambda_m$, so that every neighbor of $\partial\Lambda_m$ is revealed before $\tau$. Hence, $C$ is not connected by a white path to $\partial\Lambda_m$ and it follows that $\partial C$ is a circuit circling around $\Lambda_k$, which we call $\Gamma$. By the second part of Lemma 2.1, $\omega^{(m)} \ge \omega$ and $\omega^{(n)} \ge \omega$ so that every site in $\Gamma$ is black as well in $\omega^{(m)}$ and $\omega^{(n)}$.

We now prove by induction on $i \ge \tau$ that for all $x \in X_{[i]} \cap C$, $\omega_x^{(m)} = \omega_x^{(n)}$. The set $X_{[i]} \cap C$ is indeed empty at $i = \tau$, and for $i \ge \tau + 1$, by the first equality in Lemma 2.2,

$$ \mathbf{P}(\varepsilon_{X(i)} = 1 | \varepsilon_{X_{[i-1]}} = \omega_{X_{[i-1]}}^{(m)}, 0 \leftrightarrow \partial\Lambda_m) = \mathbf{P}(\varepsilon_{X(i)} = 1 | \varepsilon_{X_{[i-1]}} = \omega_{X_{[i-1]}}^{(m)}, 0 \leftrightarrow \Gamma), $$

keeping in mind that $X$, $\omega^{(m)}$ and $\Gamma$ are treated as deterministic when using $\mathbf{P}$. By the second equality in Lemma 2.2, we can replace the conditioning on $\{\varepsilon_{X_{[i-1]}} = \omega_{X_{[i-1]}}^{(m)}\}$ by a conditioning on $\{\varepsilon_{X_{[i-1]} \cap C} = \omega_{X_{[i-1]} \cap C}^{(m)}\}$. Under the induction hypothesis that $\omega^{(n)} = \omega^{(m)}$ on $X_{[i-1]} \cap C$, we deduce

$$ \mathbf{P}(\varepsilon_{X(i)} = 1 | \varepsilon_{X_{[i-1]}} = \omega_{X_{[i-1]}}^{(m)}, 0 \leftrightarrow \partial\Lambda_m) = \mathbf{P}(\varepsilon_{X(i)} = 1 | \varepsilon_{X_{[i-1]}} = \omega_{X_{[i-1]}}^{(n)}, 0 \leftrightarrow \partial\Lambda_n). $$

By (3), we deduce that if $\omega^{(n)} = \omega^{(m)}$ on $X_{[i-1]} \cap C$ and if $X(i) \in C$ then $\omega_{X(i)}^{(n)} = \omega_{X(i)}^{(m)}$, which concludes the induction. Therefore, on the event $\{X_{[\tau]} \cap \Lambda_k = \emptyset\}$, $\omega^{(m)}$ and $\omega^{(n)}$ coincide in $\Lambda_k$, so that

$$ \{\omega_{\Lambda_k}^{(m)} \neq \omega_{\Lambda_k}^{(n)}\} \subseteq \{X_{[\tau]} \cap \Lambda_k \neq \emptyset\} \stackrel{(4)}{=} \{\Lambda_k \leftrightarrow^* \partial\Lambda_n\}. $$

Finally, if $E$ is an event depending only on $\Lambda_k$,

$$ \left| \mathbf{P}(E | 0 \leftrightarrow \partial\Lambda_m) - \mathbf{P}(E | 0 \leftrightarrow \partial\Lambda_n) \right| \le \mathbb{P}(\omega_{\Lambda_k}^{(m)} \neq \omega_{\Lambda_k}^{(n)}) \le \mathbb{P}(\Lambda_k \leftrightarrow^* \partial\Lambda_n). \quad \square $$

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Malo Hillairet:

Institut Fourier, UMR 5582, Laboratoire de Mathématiques, Université Grenoble Alpes, CS 40700,
38058 Grenoble cedex 9, France

Email: malo.hillairet@univ-grenoble-alpes.fr

Url: https://www-fourier.univ-grenoble-alpes.fr/ hillairm/