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Sharp relations between volume growth, isoperimetry and escape probability in vertex-transitive graphs

Romain Tessera, Matthew Tointon

Abstract

We prove sharp bounds on the probability that the simple random walk on a vertex-transitive graph escapes the ball of radius $r$ before returning to its starting point. In particular, this shows that if the ball of radius $r$ has size slightly greater than quadratic in $r$ then this probability is bounded from below. On the other hand, we show that if the ball of radius $r$ has volume slightly less than cubic in $r$ then this probability decays logarithmically for all larger balls. These results represent a finitary refinement of Varopoulos's theorem that a random walk on a vertex-transitive graph is recurrent if and only if the graph has at most quadratic volume growth. They also imply the existence of a gap at $0$ for escape probabilities: there exists a universal constant $c>0$ such that the random walk on an arbitrary vertex-transitive graph is either recurrent or has a probability of at least $c$ of escaping to infinity. We also prove versions of these results for finite graphs, in particular confirming and strengthening a conjecture of Benjamini and Kozma from 2002. Amongst other things, we also generalise our results to give a sharp finitary version of the characterisation of $p$-parabolic vertex-transitive graphs, prove a number of sharp isoperimetric inequalities for vertex-transitive graphs, and prove a locality result for the escape probability of the random walk on a vertex-transitive graph that can be seen as an analogue of Schramm's locality conjecture for the critical percolation probability.

Sharp relations between volume growth, isoperimetry and escape probability in vertex-transitive graphs

Abstract

We prove sharp bounds on the probability that the simple random walk on a vertex-transitive graph escapes the ball of radius before returning to its starting point. In particular, this shows that if the ball of radius has size slightly greater than quadratic in then this probability is bounded from below. On the other hand, we show that if the ball of radius has volume slightly less than cubic in then this probability decays logarithmically for all larger balls. These results represent a finitary refinement of Varopoulos's theorem that a random walk on a vertex-transitive graph is recurrent if and only if the graph has at most quadratic volume growth. They also imply the existence of a gap at for escape probabilities: there exists a universal constant such that the random walk on an arbitrary vertex-transitive graph is either recurrent or has a probability of at least of escaping to infinity. We also prove versions of these results for finite graphs, in particular confirming and strengthening a conjecture of Benjamini and Kozma from 2002. Amongst other things, we also generalise our results to give a sharp finitary version of the characterisation of -parabolic vertex-transitive graphs, prove a number of sharp isoperimetric inequalities for vertex-transitive graphs, and prove a locality result for the escape probability of the random walk on a vertex-transitive graph that can be seen as an analogue of Schramm's locality conjecture for the critical percolation probability.

Paper Structure

This paper contains 14 sections, 59 theorems, 173 equations, 1 table.

Table of Contents

  1. Introduction
  2. Statements (and generalisations) of results in terms of resistance
  3. Electric resistance
  4. Generalisation to $p$-parabolicity
  5. Benjamini and Kozma's resistance conjecture
  6. Background on $p$-resistance
  7. An isoperimetric bound on $p$-resistance
  8. Isoperimetric inequalities from lower bounds on growth
  9. Growth bounds
  10. Isoperimetric inequalities in large balls
  11. Upper bounds on $p$-resistance
  12. The Nash-Williams inequality and lower bounds on $p$-resistance
  13. Tightness of the bounds in our main theorems
  14. Locality of the escape probability

Key Result

Theorem 1.1

Let $\Gamma$ be a connected, locally finite vertex-transitive graph, and suppose that $\beta_\Gamma(r)/r^2\to\infty$ as $r\to\infty$. Then $\mathbb{P}[\,x\to\infty\,]>0$ for every $x\in\Gamma$.

Theorems & Definitions (111)

  • Theorem 1.1: Varopoulos
  • Theorem 1.2
  • Corollary 1.3: gap at $0$ for the escape probability on vertex-transitive graphs
  • Theorem 1.4
  • Example 1.5
  • Theorem 1.6
  • Corollary 1.7
  • proof
  • Theorem 1.8
  • Theorem 1.9
  • ...and 101 more