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

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.