Existence and properties of connections decay rate for high temperature percolation models

Let $P$ denote an edge-percolation measure on $\mathbb{Z}^{d}$. The central object of our investigation is(are) the rate(s) of exponential decay for point-to-point connection probabilities (two point functions):

The main motivation of this work comes from the (supposed) universal behaviour of two point functions in high temperature systems: they should decay exponentially with a well-defined rate and the pre-factor to this decay should be the one predicted by the Ornstein-Zernike theory [10, 14]. See [13] for a review on this topic.

On the one hand some fairly satisfactory universal statements are available in perturbative regimes (very high temperature regime), see [2]. In the other hand, a non-perturbative approach (giving statements about the whole high temperature regime) has been developed over the past decades, proving the expected behaviour in various specific models: [1, 7, 8, 4, 5, 6, 11]. A recurrent ingredient in the proofs being the presence of correlation inequalities.

The latest non-perturbative approaches (mainly [6] combined with refinements from [12] and [11]) seem to be robust enough to tackle the problem (with some work…) for generic percolation models under a high temperature assumption and conditionally on the decay rate existence as well as some of its properties validity. This latter condition is usually where correlation inequalities are crucially needed.

To give an idea of the problem, let us consider some translation invariant percolation model $P$. When $P$ satisfies the FKG inequality, one has $P(0\leftrightarrow x+y)\geq P(0\leftrightarrow x\leftrightarrow x+y)\geq P(0\leftrightarrow x)P(0\leftrightarrow y)$. The equality $\overline{\nu}=\underline{\nu}\equiv\nu$ is then easy consequence of Fekete’s Lemma. One can further extend $\nu$ by positive homogeneity. The above inequality directly implies that $\nu$ satisfies the triangle inequality.

The main problem is that “satisfying FKG inequality” (or any other) is a very model specific property (which fails for some arbitrarily small perturbation of -for example- FK percolation) while $\overline{\nu}=\underline{\nu}$ and $\nu$ being a norm should be a generic property of high temperature models (which is a condition insensitive to sufficiently small perturbations). We therefore introduce a suitable notion of high temperature phase for percolation models and prove that the wanted properties of $\nu$ hold for any model in this phase. To the best of the author’s knowledge, this is the first non-perturbative proof of this type of result not relying on correlation inequalities.

Our main result is (see Section 2 for missing definitions):

The idea of the proof goes as follows: one expects that existence of $\nu$ and $\nu$ being a norm is closely related to some form of sub-additivity. The latter property can be recovered from mixing if typical clusters realizing connections are somehow directed. We thus introduce various notions of “directed connections” for which we prove existence of an asymptotic decay rate. We then show that all these rates are equal and define a norm $\tilde{\nu}$. To relate the obtained “directed rate” to the “real rates”, we do a small detour: we introduce point to hyperplanes decay rates and their directed version. Showing that these two agree is much easier than for point-to-point connections and is done using a suitable coarse-graining argument. We then relate directed point-to-point to directed point-to-hyperplane via convex duality (approximately: directed point-to-hyperplane connections in a direction $s$ are realized by a directed point-to-point connection in an optimal direction $s^{\prime}$). Finally, we relate (non directed) point-to-point connections to (non directed) point-to-hyperplane connections via another coarse-graining argument.

Denote $\|\ \|$ the Euclidean norm on $\mathbb{R}^{d}$ and $\mathrm{d}$ the associated distance. Write $\mathbb{S}^{d-1}$ the unit sphere for $\|\ \|$. $\langle\ ,\ \rangle$ will denote the scalar product. $s$ will always be an element of $\mathbb{S}^{d-1}$. For a (possibly asymmetric) norm $\mu:\mathbb{R}^{d}\to\mathbb{R}_{+}$, define the unite ball of $\mu$ and its polar set (“Wulff shape”)

For $A\subset\mathbb{R}^{d}$ and $x\in\mathbb{R}^{d}$, write $A+x$ the translate of $A$ by $x$, $\partial A$ the boundary of $A$ and $\mathring{A}=A\setminus\partial A$ the interior of $A$.

Denote

We also denote $\Lambda_{N}$ the intersection of $\Lambda_{N}$ with $\mathbb{Z}^{d}$.

Define the half spaces: for $s\in\mathbb{S}^{d-1}$,

Then, for $\delta\in[0,1]$, define the cones

$\delta=0$ is a line and $\delta=1$ is the half space $H_{s}$.

Also introduce the truncated cones

For $x\in\mathbb{R}^{d}$, we denote $\mathrm{int}(x)$ the point in $\mathbb{Z}^{d}$ closest to $x$, with some fixed breaking of draws respecting symmetries/translations of $\mathbb{Z}^{d}$. We will often omit $\mathrm{int}$ from the notation.

We fix a priori some arbitrary total order on $\mathbb{Z}^{d}$.

We will regularly use the following notation: for $(a_{n})_{n\geq 1}\in\mathbb{R}^{\mathbb{N}}$ a sequence, we denote $\overline{a}=\limsup_{n\to\infty}a_{n}\in\mathbb{R}\cup\{\pm\infty\}$, $\underline{a}=\liminf_{n\to\infty}a_{n}\in\mathbb{R}\cup\{\pm\infty\}$. When $\overline{a}=\underline{a}$, we write the limit $a$.

A quantity $f(n)$ is $o_{n}(1)$ if $\lim_{n\to\infty}f(n)=0$.

We consider edge percolation models, in all this work $E$ will be a subset of $\big{\{}\{i,j\}\subset\mathbb{Z}^{d}\big{\}}$ with the properties:

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  Irreducibility: $(\mathbb{Z}^{d},E)$ is connected.

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  Finite Range: there exists $r>0$ such that $\|i-j\|\geq r\implies\{i,j\}\notin E$. The smallest such $r$ is called the range of $E$ and is denoted $R\equiv R(E)$.

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  Translation Invariance: for any $e\in E$ and $x\in\mathbb{Z}^{d}$, $x+e\in E$.

The graph distance on $(\mathbb{Z}^{d},E)$ is denoted $\mathrm{d}_{E}$. As $E$ is finite range and irreducible, there exists $c_{E}>0$ such that

For a set $A\subset\mathbb{Z}^{d}$, denote $A^{c}=\mathbb{Z}^{d}\setminus A$, $\partial^{\mathrm{\scriptscriptstyle int}}A=\{x\in A:\ \exists y\in A^{c},\ \{x,y\}\in E\}$, $\partial^{\mathrm{\scriptscriptstyle ext}}A=\{x\in A^{c}:\ \exists y\in A,\ \{x,y\}\in E\}$. Also define $E(A)=\{\{x,y\}\in E:\ x,y\in A\}$.

For $\omega\in\{0,1\}^{E}$, we systematically identify the $\{0,1\}$-valued function and the edge set induced by the set $\{e:\omega_{e}=1\}$, the set of open edges. When talking about connectivity properties of $\omega$, it is assumed that the graph $(\mathbb{Z}^{d},\omega)$ is considered.

For $F\subset E$ finite, denote $\mathcal{F}_{F}=\{A\subset\{0,1\}^{F}\}$ and for $F\subset E$ infinite, denote $\mathcal{F}_{F}$ the sigma algebra generated by the collection $(\mathcal{F}_{F^{\prime}})_{F^{\prime}\subset F\textnormal{ finite}}$. A percolation measure $P$ is a probability measure such that $(P,\mathcal{F}_{E},E)$ is a probability space. We write $\{x\leftrightarrow y\}$ for the event that $x,y$ lie in the same connected component (and $\{A\leftrightarrow B\}$ for the event that there exists $x\in A,y\in B$ with $x\leftrightarrow y$). We also will write $\{x\xleftrightarrow{F}y\}$ for the event that $x$ is connected to $y$ by a path of open edges in $F$. $\omega$ will be a random variable with law $P$.

One of our hypotheses is a mixing condition, called the exponential ratio weak mixing property for connections events:

A useful consequence of insertion tolerance is

We will regularly use this kind of argument without explicitly writing down the details.

To avoid dealing with trivialities occurring from the discrete structure of $\mathbb{Z}^{d}$, we will look at a coarser notion of connections. Let $R_{0}\geq R$ (recall $R$ is the range) be a fixed integer such that $\big{(}\Lambda_{R_{0}},E(\Lambda_{R_{0}})\big{)}$ is connected. Denote $\Gamma=((2R_{0}+1)\mathbb{Z})^{d}$ the coarser lattice. For $x\in\Gamma$, denote $\Lambda(x)=x+\Lambda_{R_{0}}$. To lighten notations, for $x,y\in\Gamma$ we will write $\{x\leftrightarrow y\}$ for the event $\{\Lambda(x)\leftrightarrow\Lambda(y)\}$. By Lemma 2.1, these events have the same asymptotic decay rates as the point-to-point rates.

For a point $x\in\mathbb{R}^{d}$, denote $B_{x}=v_{x}+[-R_{0}-1/2,R_{0}+1/2)^{d}$ the box such that $v_{x}\in\Gamma$, $x\in B_{x}$. For a set $\Delta\subset\mathbb{R}^{d}$, we denote $[\Delta]=\bigcup_{x\in\Delta}B_{x}\cap\mathbb{Z}^{d}$. For $x,y\in\mathbb{R}^{d},\Delta\subset\mathbb{R}^{d}$, we write $\{x\xleftrightarrow{\Delta}y\}=\{\Lambda(v_{x})\xleftrightarrow{E([\Delta])}\Lambda(v_{y})\}$.

In the same spirit, for $\Delta\subset\mathbb{R}^{d}$, we say that an event is $\Delta$-measurable if it is in $\mathcal{F}_{E([\Delta])}$.

We will regularly use coarse-graining of the cluster of $0$. We describe here a generic coarse-graining procedure parametrized by the “unit cell” of the coarse graining. These procedures are a general formulation of the coarse-graining procedure applied in [6]. Let $0\in\Delta\subset\mathbb{Z}^{d}$ be finite. Let $\Delta_{K}=\bigcup_{x\in\Delta}\Lambda_{K}(x)$. Let $\mathcal{T}=\mathcal{T}(\Delta,K)$ be the set of embedded rooted trees defined as follows: $T\in\mathcal{T}$ is the data of a set of vertices $t=\{t_{0},\cdots,t_{m}\}$ where each $t_{i}\in\mathbb{Z}^{d}$, and a set of edges $f=\{f_{1},\cdots,f_{m}\}$ with $f_{i}\subset t,|f_{i}|=2$ such that

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  The graph $(t,f)$ is a tree.

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  A given point in $\mathbb{Z}^{d}$ can only occur once as element of $t$.

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  $t_{0}=0$, $t_{i}\in\partial^{\mathrm{\scriptscriptstyle ext}}(\Delta_{K}+t_{j})$ where $f_{i}=\{t_{i},t_{j}\}$.

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  The labels and edges can be inductively reconstructed from the set of vertices (without labels) $W$ as follows: $t_{i}$ is the smallest (for the fixed total order on $\mathbb{Z}^{d}$) element of $W\setminus\{0,t_{1},\cdots,t_{i-1}\}$ belonging to $\bigcup_{j=0}^{i-1}\partial^{\mathrm{\scriptscriptstyle ext}}(t_{j}+\Delta_{K})$ and $f_{i}$ is given by $\{t_{i},v^{*}\}$ where $v^{*}$ is the smallest element of $\{t_{0},\cdots,t_{i-1}\}$ with $t_{i}\in\partial^{\mathrm{\scriptscriptstyle ext}}(v^{*}+\Delta_{K})$.

A fairly direct observation is that the degree of a vertex $t_{i}$ in $(t,f)$ is less than $d_{\Delta_{K}}=|\partial^{\mathrm{\scriptscriptstyle ext}}\Delta_{K}|$ and one has a natural inclusion of $\mathcal{T}_{l}=\{T\in\mathcal{T}:\ |t|=l\}$ in the set of sub-trees of $\mathbb{T}_{d_{\Delta_{K}}}$ (the $d_{\Delta_{K}}$-regular tree) containing $0$ and having $l$ vertices. In particular, there exists $c>0$ universal such that $|\mathcal{T}_{l}|\leq e^{c\log(d_{\Delta_{K}})l}$.

We now define a mapping $\mathrm{CG}_{\Delta,K}$ from the set of clusters containing $0$ to $\mathcal{T}(\Delta,K)$. We define it via an algorithm constructing $T\in\mathcal{T}$ from $C\ni 0$ (see Figure 1). Fix some $C\ni 0$. Consider the graph formed by the vertices of $\mathbb{Z}^{d}$ and the edges in $C$. Construct $t,f$ as follows

Write $\mathrm{CG}_{\Delta,K}(C)=(t(C),f(C))$. One has automatically that $C$ is in a $(K+2\times\mathrm{radius}(\Delta))$-neighbourhood of $\mathrm{CG}_{\Delta,K}(C)$.

The usefulness of such coarse graining is the conjunction of the combinatorial control we mentioned on trees with given number of vertices and the following energy bound.

First introduce a family of connection events. For $\delta\in(0,1]$ and $s,s^{\prime}\in\mathbb{S}^{d-1}$ such that $s\in\mathring{\mathcal{Y}}_{s^{\prime},\delta}$,

From Lemma 4.1 and 4.2, it is natural to introduce $\tilde{\nu}:\mathbb{R}^{d}\to\mathbb{R}_{+}$ as the extension by positive homogeneity of $\tilde{\nu}_{s^{\prime},\delta}(s)$.