论文标题
在顶点传输图中的体积增长，等术和逃逸概率之间的急剧关系
Sharp relations between volume growth, isoperimetry and escape probability in vertex-transitive graphs
论文作者
论文摘要
我们证明，在返回其起点之前，简单的随机步行图逃脱了半径$ r $的球的概率。特别是，这表明，如果半径$ r $的球的大小略大于$ r $中的二次大小，则此概率从下面有限。另一方面，我们表明，如果半径$ r $的球的体积略低于$ r $的立方体，那么对于所有较大的球而言，这种概率会衰减。这些结果代表了Varopoulos定理的一个细化，即当且仅当图形最多具有二次体积增长时，在顶点传递图上随机行走是经常出现的。它们还意味着逃逸概率的差距为$ 0 $：存在通用常数$ c> 0 $，以便在任意顶点传递图上随机步行要么是反复出现的，要么具有至少$ c $逃逸到无限限的概率。 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在顶点传递图上随机行走的概率可以看作是Schramm的位置猜想的类似物，以构成关键渗透概率。
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.
