A new proof of the sharpness of the phase transition for Bernoulli percolation on \mathbb{Z}^{d}

A new proof of the sharpness of the phase transition for Bernoulli percolation on \mathbb{Z}^{d}

Hugo Duminil-Copin and Vincent Tassion
July 8, 2019
Abstract

We provide a new proof of the sharpness of the phase transition for nearest-neighbour Bernoulli percolation. More precisely, we show that

  • for p<p_{c}, the probability that the origin is connected by an open path to distance n decays exponentially fast in n.

  • for p>p_{c}, the probability that the origin belongs to an infinite cluster satisfies the mean-field lower bound \theta(p)\ge\tfrac{p-p_{c}}{p(1-p_{c})}.

This note presents the argument of [DCT15], which is valid for long-range Bernoulli percolation (and for the Ising model) on arbitrary transitive graphs in the simpler framework of nearest-neighbour Bernoulli percolation on \mathbb{Z}^{d}.

1 Statement of the result

Notation.

Fix an integer d\ge 2. We consider the d-dimensional hypercubic lattice (\mathbb{Z}^{d},\mathbb{E}^{d}). Let \Lambda_{n}=\{-n,\ldots,n\}^{d}, and let \partial\Lambda_{n}:=\Lambda_{n}\setminus\Lambda_{n-1} be its vertex-boundary. Throughout this note, S always stands for a finite set of vertices containing the origin. Given such a set, we denote its edge-boundary by \Delta S, defined by all the edges \{x,y\} with x\in S and y\notin S.

Consider the Bernoulli bond percolation measure \mathbb{P}_{p} on \{0,1\}^{\mathbb{E}^{d}} for which each edge of \mathbb{E}^{d} is declared open with probability p and closed otherwise, independently for different edges.

Two vertices x and y are connected in S\subset V if there exists a path of vertices (v_{k})_{0\le k\le K} in S such that v_{0}=x, v_{K}=y, and \{v_{k},v_{k+1}\} is open for every 0\le k<K. We denote this event by x\lx@stackrel{{\scriptstyle S}}{{\leftarrow\mathrel{\mskip-3.1mu }\rightarrow}}y. If S=\mathbb{Z}^{d}, we drop it from the notation. We set 0\lx@stackrel{{\scriptstyle}}{{\leftarrow\mathrel{\mskip-3.1mu }\rightarrow}}\infty (resp. 0\lx@stackrel{{\scriptstyle}}{{\leftarrow\mathrel{\mskip-3.1mu }\rightarrow}}% \partial\Lambda_{n}) if 0 is connected to infinity (resp. 0 is connected to a vertex in \partial\Lambda_{n}).

Phase transition.

A new idea of this paper is to use a different definition of the critical parameter than the standard one. This new definition relies on the following quantity. For p\in[0,1] and 0\in S\subset\mathbb{Z}^{d}, define

\varphi_{p}(S):=p\tsum\slimits@_{\{x,y\}\in\Delta S}\mathbb{P}_{p}[0% \lx@stackrel{{\scriptstyle S}}{{\leftarrow\mathrel{\mskip-3.1mu }\rightarrow}}x] (1.1)

and introduce the following quantities:

\displaystyle\tilde{p}_{c} \displaystyle:=\sup\big{\{}p\in[0,1]\text{ s.t.\@ there exists a finite set $0\subset S\subset\mathbb{Z}^{d}$ with $\varphi_{p}(S)<1$}\big{\}}, (1.2)
\displaystyle p_{c} \displaystyle:=\sup\{p\text{ s.t. $\mathbb{P}_{p}[0\leftarrow\mathrel{\mskip-3% .1mu }\rightarrow\infty]=0$}\}.

We are now in a position to state our main result.

Theorem 1.1.

For any d\ge 2, \tilde{p}_{c}=p_{c}. Furthermore,

  1. For p<p_{c}, there exists c=c(p)>0 such that for every n\ge 1,

    \mathbb{P}_{p}[0\leftarrow\mathrel{\mskip-3.1mu }\rightarrow\partial\Lambda_{n% }]\le e^{-cn}.
  2. For p>p_{c},

    \mathbb{P}_{p}[0\leftarrow\mathrel{\mskip-3.1mu }\rightarrow\infty]\ge\frac{p-% p_{c}}{p(1-p_{c})}.

Remarks.

  1. We refer to [DCT15] for a detailed bibliography, and for a version of the proof valid in greater generality. The aim of this note is to provide a proof in the simplest framework.

  2. Theorem 1.1 was proved by Aizenman and Barsky [AB87] in the more general framework of long-range percolation. In their proof, they consider an additional parameter h corresponding to an external field, and they derive the results from differential inequalities satisfied by the thermodynamical quantities of the model. A different proof, based on the geometric study of the pivotal edges, was obtained at the same time by Menshikov [Men86]. These two proofs are also presented in [Gri99].

  3. In the definition of \tilde{p}_{c}, the set of parameters p such that there exists a finite set 0\subset S\subset\mathbb{Z}^{d} with \varphi_{p}(S)<1 is an open subset of [0,1]. Thus, \tilde{p}_{c} do not belong to this set.

    We obtain that the expected size of the cluster of the origin satisfies that for every p>p_{c},

    \tsum\slimits@_{x\in\mathbb{Z}^{d}}\mathbb{P}_{p}[0\leftarrow\mathrel{\mskip-3% .1mu }\rightarrow x]\ge\tsum\slimits@_{n\ge 0}\varphi_{p}(\Lambda_{n})=+\infty.
  4. Since \varphi_{p}(\{0\})=2dp, we obtain p_{c}\ge 1/{2d}.

  5. Item 2 provides a mean-field lower bound for the infinite cluster density.

  6. Theorem 1.1 implies that p_{c}\le 1/2 on \mathbb{Z}^{2}. Combined with Zhang’s argument [Gri99, Lemma 11.12], this shows that p_{c}=1/2.

2 Proof of the theorem

It is sufficient to show Items 1 and 2 with p_{c} replaced by \tilde{p}_{c} (since it immediately implies the equality p_{c}=\tilde{p}_{c}).

2.1 Proof of Item 1

The proof of Item 1 can be derived from the BK-inequality [vdBK85]. We present here an exploration argument, similar to the one in [Ham57], which does not rely on the BK-inequality. Let p<\tilde{p}_{c}. By definition, one can fix a finite set S containing the origin, such that \varphi_{p}(S)<1. Let L>0 such that S\subset\Lambda_{L-1}.

Let k\ge 1 and assume that the event 0\lx@stackrel{{\scriptstyle}}{{\leftarrow\mathrel{\mskip-3.1mu }\rightarrow}}% \partial\Lambda_{kL} holds. Let

\mathrsfs{C}=\{z\in S:0\lx@stackrel{{\scriptstyle S}}{{\leftarrow\mathrel{% \mskip-3.1mu }\rightarrow}}z\}.

Since S\cap\partial\Lambda_{kL}=\emptyset, there exists an edge \{x,y\}\in\Delta S such that the following events occur:

  • 0 is connected to x in S,

  • \{x,y\} is open,

  • y is connected to \partial\Lambda_{kL} in \mathrsfs{C}^{c}.

Using first the union bound, and then a decomposition with respect to possible values of \mathrsfs{C}, we find

\displaystyle\mathbb{P}_{p}[0\lx@stackrel{{\scriptstyle}}{{\leftarrow\mathrel{% \mskip-3.1mu }\rightarrow}}\partial\Lambda_{kL}] (2.1)
\displaystyle\le\tsum\slimits@_{\{x,y\}\in\Delta S}\tsum\slimits@_{C\subset S}% \mathbb{P}_{p}\big{[}\{0\lx@stackrel{{\scriptstyle S}}{{\leftarrow\mathrel{% \mskip-3.1mu }\rightarrow}}x,\mathrsfs{C}=C\}\cap\{\{x,y\}\text{ is open}\}% \cap\{y\lx@stackrel{{\scriptstyle\mathbb{Z}^{d}\setminus C}}{{\leftarrow% \mathrel{\mskip-3.1mu }\rightarrow}}\partial\Lambda_{kL}\}\big{]} (2.2)
\displaystyle=p\tsum\slimits@_{\{x,y\}\in\Delta S}\tsum\slimits@_{C\subset S}% \mathbb{P}_{p}\big{[}0\lx@stackrel{{\scriptstyle S}}{{\leftarrow\mathrel{% \mskip-3.1mu }\rightarrow}}x,\mathrsfs{C}=C\big{]}\mathbb{P}_{p}\big{[}y% \lx@stackrel{{\scriptstyle\mathbb{Z}^{d}\setminus C}}{{\leftarrow\mathrel{% \mskip-3.1mu }\rightarrow}}\partial\Lambda_{kL}\big{]}. (2.3)

In the second line, we used the fact that the three events depend on different sets of edges and are therefore independent. Since y\in\Lambda_{L}, one can bound \mathbb{P}_{p}[y\lx@stackrel{{\scriptstyle\mathbb{Z}^{d}\setminus C}}{{% \leftarrow\mathrel{\mskip-3.1mu }\rightarrow}}\partial\Lambda_{kL}] by \mathbb{P}_{p}[0\lx@stackrel{{\scriptstyle}}{{\leftarrow\mathrel{\mskip-3.1mu % }\rightarrow}}\partial\Lambda_{(k-1)L}] in the last expression. Hence, we find

\mathbb{P}_{p}[0\lx@stackrel{{\scriptstyle}}{{\leftarrow\mathrel{\mskip-3.1mu % }\rightarrow}}\partial\Lambda_{kL}]\le\varphi_{p}(S)\mathbb{P}_{p}\big{[}y% \lx@stackrel{{\scriptstyle}}{{\leftarrow\mathrel{\mskip-3.1mu }\rightarrow}}% \partial\Lambda_{(k-1)L}\big{]} (2.4)

which by induction gives

\mathbb{P}_{p}[0\lx@stackrel{{\scriptstyle}}{{\leftarrow\mathrel{\mskip-3.1mu % }\rightarrow}}\partial\Lambda_{kL}]\le\varphi_{p}(S)^{k-1}. (2.5)

This proves the desired exponential decay.

2.2 Proof of Item 2

Let us start by the following lemma providing a differential inequality valid for every p.

Lemma 2.1.

Let p\in[0,1] and n\ge 1,

\frac{d}{dp}\mathbb{P}_{p}[0\lx@stackrel{{\scriptstyle}}{{\leftarrow\mathrel{% \mskip-3.1mu }\rightarrow}}\partial\Lambda_{n}]\ge\frac{1}{p(1-p)}\cdot\inf_{% \begin{subarray}{c}S\subset\Lambda_{n}\\ 0\in S\end{subarray}}\varphi_{p}(S)\cdot\big{(}1-\mathbb{P}_{p}[0\lx@stackrel{% {\scriptstyle}}{{\leftarrow\mathrel{\mskip-3.1mu }\rightarrow}}\partial\Lambda% _{n}]\big{)}. (2.6)

Let us first see how it implies Item 2 of Theorem 1.1. Integrating the differential inequality (2.6) between \tilde{p}_{c} and p>\tilde{p}_{c} implies that for every n\ge 1, \mathbb{P}_{p}[0\lx@stackrel{{\scriptstyle}}{{\leftarrow\mathrel{\mskip-3.1mu % }\rightarrow}}\partial\Lambda_{n}]\ge\frac{p-\tilde{p}_{c}}{p(1-\tilde{p}_{c})}. By letting n tend to infinity, we obtain the desired lower bound on \mathbb{P}_{p}[0\leftarrow\mathrel{\mskip-3.1mu }\rightarrow\infty].

Proof of Lemma 2.1.

Recall that \{x,y\} is pivotal for the configuration \omega and the event \{0\leftarrow\mathrel{\mskip-3.1mu }\rightarrow\partial\Lambda_{n}\} if \omega_{\{x,y\}}\notin\{0\leftarrow\mathrel{\mskip-3.1mu }\rightarrow\partial% \Lambda_{n}\} and \omega^{\{x,y\}}\in\{0\leftarrow\mathrel{\mskip-3.1mu }\rightarrow\partial% \Lambda_{n}\}. (The configuration \omega_{\{x,y\}}, resp. \omega^{\{x,y\}}, coincides with \omega except that the edge \{x,y\} is closed, resp. open.) By Russo’s formula (see [Gri99, Section 2.4]), we have

\displaystyle\frac{d}{dp}\mathbb{P}_{p}[0\lx@stackrel{{\scriptstyle}}{{% \leftarrow\mathrel{\mskip-3.1mu }\rightarrow}}\partial\Lambda_{n}] \displaystyle=\tsum\slimits@_{e\subset\Lambda_{n}}\mathbb{P}_{p}\big{[}\text{$% e$ is pivotal}\big{]} (2.7)
\displaystyle=\frac{1}{1-p}\tsum\slimits@_{e\subset\Lambda_{n}}\mathbb{P}_{p}% \big{[}\text{$e$ is pivotal},\,0\overset{}{\mathrel{\mathchoice{\ooalign{$% \hfil\displaystyle/\hfil$\cr$\displaystyle\leftarrow\mathrel{\mskip-3.1mu }% \rightarrow$}}{\ooalign{$\hfil\textstyle/\hfil$\cr$\textstyle\leftarrow% \mathrel{\mskip-3.1mu }\rightarrow$}}{\ooalign{$\hfil\scriptstyle/\hfil$\cr$% \scriptstyle\leftarrow\mathrel{\mskip-3.1mu }\rightarrow$}}{\ooalign{$\hfil% \scriptscriptstyle/\hfil$\cr$\scriptscriptstyle\leftarrow\mathrel{\mskip-3.1mu% }\rightarrow$}}}}\partial\Lambda_{n}\big{]}. (2.8)

Define the following random subset of \Lambda_{n}:

\mathrsfs{S}:=\{x\in\Lambda_{n}\text{ such that }x\mathrel{\mathchoice{% \ooalign{$\hfil\displaystyle/\hfil$\cr$\displaystyle\leftarrow\mathrel{\mskip-% 3.1mu }\rightarrow$}}{\ooalign{$\hfil\textstyle/\hfil$\cr$\textstyle\leftarrow% \mathrel{\mskip-3.1mu }\rightarrow$}}{\ooalign{$\hfil\scriptstyle/\hfil$\cr$% \scriptstyle\leftarrow\mathrel{\mskip-3.1mu }\rightarrow$}}{\ooalign{$\hfil% \scriptscriptstyle/\hfil$\cr$\scriptscriptstyle\leftarrow\mathrel{\mskip-3.1mu% }\rightarrow$}}}\partial\Lambda_{n}\}.

The boundary of \mathrsfs{S} corresponds to the outmost blocking surface (which can be obtained by exploring from the outside the set of vertices connected to the boundary). When 0 is not connected to \partial\Lambda_{n}, the set \mathrsfs{S} is always a subset of \Lambda_{n} containing the origin. By summing over the possible values for \mathrsfs{S}, we obtain

\frac{d}{dp}\mathbb{P}_{p}[0\lx@stackrel{{\scriptstyle}}{{\leftarrow\mathrel{% \mskip-3.1mu }\rightarrow}}\partial\Lambda_{n}]=\frac{1}{1-p}\tsum\slimits@_{% \begin{subarray}{c}S\subset\Lambda_{n}\\ 0\in S\end{subarray}}\tsum\slimits@_{e\subset\Lambda_{n}}\mathbb{P}_{p}\big{[}% \text{$e$ is pivotal},\,\mathrsfs{S}=S\big{]} (2.9)

Observe that on the event \mathrsfs{S}=S, the pivotal edges are the edges \{x,y\}\in\Delta S such that 0 is connected to x in S. This implies that

\frac{d}{dp}\mathbb{P}_{p}[0\lx@stackrel{{\scriptstyle}}{{\leftarrow\mathrel{% \mskip-3.1mu }\rightarrow}}\partial\Lambda_{n}]=\frac{1}{1-p}\tsum\slimits@_{% \begin{subarray}{c}S\subset\Lambda_{n}\\ 0\in S\end{subarray}}\tsum\slimits@_{\{x,y\}\in\Delta S}\mathbb{P}_{p}\big{[}0% \lx@stackrel{{\scriptstyle S}}{{\leftarrow\mathrel{\mskip-3.1mu }\rightarrow}}% x,\,\mathrsfs{S}=S\big{]}. (2.10)

The event \{\mathrsfs{S}=S\} is measurable with respect to the configuration outside S and is therefore independent of \{0\lx@stackrel{{\scriptstyle S}}{{\leftarrow\mathrel{\mskip-3.1mu }% \rightarrow}}x\}. We obtain

\displaystyle\frac{d}{dp}\mathbb{P}_{p}[0\lx@stackrel{{\scriptstyle}}{{% \leftarrow\mathrel{\mskip-3.1mu }\rightarrow}}\partial\Lambda_{n}] \displaystyle=\frac{1}{1-p}\tsum\slimits@_{\begin{subarray}{c}S\subset\Lambda_% {n}\\ 0\in S\end{subarray}}\tsum\slimits@_{\{x,y\}\in\Delta S}\mathbb{P}_{p}\big{[}0% \lx@stackrel{{\scriptstyle S}}{{\leftarrow\mathrel{\mskip-3.1mu }\rightarrow}}% x\big{]}\mathbb{P}_{p}\big{[}\mathrsfs{S}=S\big{]} (2.11)
\displaystyle=\frac{1}{p(1-p)}\tsum\slimits@_{\begin{subarray}{c}S\subset% \Lambda_{n}\\ 0\in S\end{subarray}}\varphi_{p}(S)\mathbb{P}_{p}\big{[}\mathrsfs{S}=S\big{]} (2.12)
\displaystyle\ge\frac{1}{p(1-p)}\inf_{\begin{subarray}{c}S\subset\Lambda_{n}\\ 0\in S\end{subarray}}\varphi_{p}(S)\cdot\mathbb{P}_{p}\big{[}0\overset{}{% \mathrel{\mathchoice{\ooalign{$\hfil\displaystyle/\hfil$\cr$\displaystyle% \leftarrow\mathrel{\mskip-3.1mu }\rightarrow$}}{\ooalign{$\hfil\textstyle/% \hfil$\cr$\textstyle\leftarrow\mathrel{\mskip-3.1mu }\rightarrow$}}{\ooalign{$% \hfil\scriptstyle/\hfil$\cr$\scriptstyle\leftarrow\mathrel{\mskip-3.1mu }% \rightarrow$}}{\ooalign{$\hfil\scriptscriptstyle/\hfil$\cr$\scriptscriptstyle% \leftarrow\mathrel{\mskip-3.1mu }\rightarrow$}}}}\partial\Lambda_{n}], (2.13)

as desired. ∎

Acknowledgments

This work was supported by a grant from the Swiss FNS and the NCCR SwissMap also founded by the swiss NSF.

References

  • [AB87] Michael Aizenman and David J. Barsky. Sharpness of the phase transition in percolation models. Comm. Math. Phys., 108(3):489–526, 1987.
  • [DCT15] Hugo Duminil-Copin and Vincent Tassion. A new proof of the sharpness of the phase transition for Bernoulli percolation and the Ising model. preprint, 2015.
  • [Gri99] Geoffrey Grimmett. Percolation, volume 321 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, second edition, 1999.
  • [Ham57] J. M. Hammersley. Percolation processes: Lower bounds for the critical probability. Ann. Math. Statist., 28:790–795, 1957.
  • [Men86] M. V. Menshikov. Coincidence of critical points in percolation problems. Dokl. Akad. Nauk SSSR, 288(6):1308–1311, 1986.
  • [vdBK85] J. van den Berg and H. Kesten. Inequalities with applications to percolation and reliability. J. Appl. Probab., 22(3):556–569, 1985.

Département de Mathématiques Université de Genève Genève, Switzerland E-mail: hugo.duminil@unige.ch, vincent.tassion@unige.ch

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