顶点传递图中体积增长、等周性与逃逸概率之间的尖锐关系

Percolation and isoperimetry on roughly transitive graphs

In this paper we study percolation on a roughly transitive graph G with polynomial growth and isoperimetric dimension larger than one. For these graphs we are able to prove that p_c < 1, or in other words, that there exists a percolation phase. The main results of the article work for both dependent and independent percolation processes, since they are based on a quite robust renormalization technique. When G is transitive, the fact that p_c < 1 was already known before. But even in that case our proof yields some new results and it is entirely probabilistic, not involving the use of Gromov's …

Supercritical percolation on nonamenable graphs: Isoperimetry, analyticity, and exponential decay of the cluster size distribution

Let $G$ be a connected, locally finite, transitive graph, and consider Bernoulli bond percolation on $G$. We prove that if $G$ is nonamenable and $p > p_c(G)$ then there exists a positive constant $c_p$ such that \[\mathbf{P}_p(n \leq |K| < \infty) \leq e^{-c_p n}\] for every $n\geq 1$, where $K$ is the cluster of the origin. We deduce the following two corollaries: 1. Every infinite cluster in supercritical percolation on a transitive nonamenable graph has anchored expansion almost surely. This answers positively a question of Benjamini, Lyons, and Schramm (1997). 2. For transitive nonamenabl…

图的多项式增长上渗流的局部性：解析性、超临界锐度、等周性

Percolation on graphs of polynomial growth is local: analyticity, supercritical sharpness, isoperimetry

研究了多项式增长图上伯努利渗流的局部性，证明了超临界区域的解析性和锐度，以及最小割集连通性的新结果。

Slightly supercritical percolation on nonamenable graphs II: Growth and isoperimetry of infinite clusters

We study the growth and isoperimetry of infinite clusters in slightly supercritical Bernoulli bond percolation on transitive nonamenable graphs under the $L^2$ boundedness condition ($p_c<p_{2\to 2}$). Surprisingly, we find that the volume growth of infinite clusters is always purely exponential (that is, the subexponential corrections to growth are bounded) in the regime $p_c<p<p_{2\to 2}$, even when the ambient graph has unbounded corrections to exponential growth. For $p$ slightly larger than $p_c$, we establish the precise estimates \begin{align*} \mathbf{E}_p \left[ \# B_\mathrm{int}(v,r)…

加权等周不等式蕴含渗流

Weighted isoperimetry implies percolation

本文证明在满足加权等周不等式的图上，当等周常数足够大时，以权重为概率的独立边开放会导致正概率的无穷连通，并应用于解决长程渗流截断和传递图临界概率上界两个猜想。

计数最小顶点割集及顶点传递图上点渗流在1处的间隙

Counting Minimal Vertex Cutsets and a Gap at 1 for Site Percolation on Vertex-Transitive Graphs

本文研究顶点传递图上的点渗流，通过随机游走逃逸概率和等周型不等式给出最小顶点割集数量的指数上界，并证明临界概率在1处存在普适间隙。