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

## TL;DR

This paper proves exponential decay of cluster size distribution in supercritical percolation on nonamenable graphs, showing anchored expansion of infinite clusters and analyticity of key observables in the supercritical phase.

## Contribution

It establishes exponential decay of cluster sizes and analyticity of observables for supercritical percolation on nonamenable graphs, answering longstanding questions.

## Key findings

- Exponential decay of cluster size probabilities for p > p_c.
- Infinite clusters exhibit anchored expansion almost surely.
- Percolation probability and related functions are analytic in the supercritical phase.

## Abstract

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 nonamenable graphs, various observables including the percolation probability, the truncated susceptibility, and the truncated two-point function are analytic functions of $p$ throughout the supercritical phase.