  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.
