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Non-triviality of the phase transition for percolation on finite transitive graphs

Tom Hutchcroft, Matthew Tointon

Abstract

We prove that if $(G_n)_{n\geq1}=((V_n,E_n))_{n\geq 1}$ is a sequence of finite, vertex-transitive graphs with bounded degrees and $|V_n|\to\infty$ that is at least $(1+ε)$-dimensional for some $ε>0$ in the sense that \[\mathrm{diam} (G_n)=O\left(|V_n|^{1/(1+ε)}\right) \text{ as $n\to\infty$}\] then this sequence of graphs has a non-trivial phase transition for Bernoulli bond percolation. More precisely, we prove under these conditions that for each $0<α<1$ there exists $p_c(α)<1$ such that for each $p\geq p_c(α)$, Bernoulli-$p$ bond percolation on $G_n$ has a cluster of size at least $α|V_n|$ with probability tending to $1$ as $n\to \infty$. In fact, we prove more generally that there exists a universal constant $a$ such that the same conclusion holds whenever \[\mathrm{diam} (G_n)=O\left(\frac{|V_n|}{(\log |V_n|)^a}\right) \text{ as $n\to\infty$.}\] This verifies a conjecture of Benjamini up to the value of the constant $a$, which he suggested should be $1$. We also prove a generalization of this result to quasitransitive graph sequences with a bounded number of vertex orbits and prove that one may indeed take $a=1$ when the graphs $G_n$ are all Cayley graphs of Abelian groups. A key step in our proof is to adapt the methods of Duminil-Copin, Goswami, Raoufi, Severo, and Yadin from infinite graphs to finite graphs. This adaptation also leads to an isoperimetric criterion for infinite graphs to have a nontrivial uniqueness phase (i.e., to have $p_u<1$) which is of independent interest. We also prove that the set of possible values of the critical probability of an infinite quasitransitive graph has a gap at $1$ in the sense that for every $k,n<\infty$ there exists $ε>0$ such that every infinite graph $G$ of degree at most $k$ whose vertex set has at most $n$ orbits under Aut$(G)$ either has $p_c=1$ or $p_c\leq 1-ε$.

Non-triviality of the phase transition for percolation on finite transitive graphs

Abstract

We prove that if is a sequence of finite, vertex-transitive graphs with bounded degrees and that is at least -dimensional for some in the sense that then this sequence of graphs has a non-trivial phase transition for Bernoulli bond percolation. More precisely, we prove under these conditions that for each there exists such that for each , Bernoulli- bond percolation on has a cluster of size at least with probability tending to as . In fact, we prove more generally that there exists a universal constant such that the same conclusion holds whenever This verifies a conjecture of Benjamini up to the value of the constant , which he suggested should be . We also prove a generalization of this result to quasitransitive graph sequences with a bounded number of vertex orbits and prove that one may indeed take when the graphs are all Cayley graphs of Abelian groups. A key step in our proof is to adapt the methods of Duminil-Copin, Goswami, Raoufi, Severo, and Yadin from infinite graphs to finite graphs. This adaptation also leads to an isoperimetric criterion for infinite graphs to have a nontrivial uniqueness phase (i.e., to have ) which is of independent interest. We also prove that the set of possible values of the critical probability of an infinite quasitransitive graph has a gap at in the sense that for every there exists such that every infinite graph of degree at most whose vertex set has at most orbits under Aut either has or .

Paper Structure

This paper contains 21 sections, 53 theorems, 207 equations, 1 figure.

Table of Contents

  1. Introduction
  2. About the proof
  3. Further results
  4. Further discussion and remarks
  5. Notation
  6. Background on percolation and vertex-transitive graphs
  7. Quotients and rough isometries
  8. Quasitransitive graphs
  9. Nilpotent and Abelian groups
  10. Percolation in an elongated Euclidean box
  11. Abelian groups
  12. Finite nilpotent groups
  13. Subgroups of bounded index and virtually nilpotent groups
  14. Percolation from isoperimetry
  15. Connecting large sets via the Gaussian free field
  16. ...and 6 more sections

Key Result

Theorem 1.2

There exists an absolute constant $a\ge1$ such that for every $k,n \geq 1$, $\lambda>0$ and $\alpha,q\in(0,1)$ there exists $\varepsilon=\varepsilon(k,n,\lambda,\alpha,q)>0$ such that if $G=(V,E)$ is a finite, $n$-quasitransitive graph of degree at most $k$ satisfying then $p_c(G,\alpha,q)\le1-\varepsilon$.

Figures (1)

  • Figure 1: Left: If a square box admits both a left-right and top-bottom open crossing and a point $x$ in the box is connected to infinity in all four quarter-planes with corner at $x$, then $x$ is connected to all four sides of the box within the box. Right: Consider a rectangular box $R$ of width greater than height, let $x$ and $y$ be points in the rectangle, and let $B_1$ and $B_2$ be square boxes contained in $R$, of height equal to that of $R$, and containing $x$ and $y$ respectively. If $R$ admits an open left-right crossing and $x$ and $y$ are connected to the top and bottom of $B_1$ and $B_2$ respectively by open paths, then $x$ and $y$ are connected by an open path.

Theorems & Definitions (114)

  • Conjecture 1.1: Benjamini 2001
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4: DGRSY
  • Theorem 1.5
  • Remark 1.6
  • Theorem 1.7: Critical probability gap
  • Theorem 1.8
  • Corollary 1.9
  • Theorem 1.10
  • ...and 104 more