Supercritical long-range percolation on graphs of polynomial growth: the truncated one-arm exponent

TL;DR

This paper studies supercritical long-range percolation on graphs with polynomial growth, providing bounds on cluster radii and volume tails, and establishing a lower bound on the isoperimetric dimension of the infinite cluster.

Contribution

It offers the first bounds on cluster radius decay and volume tail behavior in supercritical long-range percolation on polynomial growth graphs, advancing understanding of geometric properties.

Findings

Bounds on the decay of finite cluster radii for supercritical percolation.

Matching order bounds on the tail volume of finite clusters.

A lower bound on the anchored isoperimetric dimension of the infinite cluster.

Abstract

We consider supercritical long-range percolation on transitive graphs of polynomial growth. In this model, any two vertices $x$ and $y$ of the underlying graph $G$ connect by a direct edge with probability $1-\exp(-\beta J(x,y))$, where $J(x,y)$ is a function that is invariant under the automorphism group of $G$, and we assume that $J$ decays polynomially with the graph distance between $x$ and $y$. We give up-to-constant bounds on the decay of the radius of finite cluster for $\beta > \beta_c$. In the same setting, we also give upper and lower bounds on the tail volume of finite clusters. The upper and lower bounds are of matching order, conjecturally on sharp volume bounds for spheres in transitive graphs of polynomial growth. As a corollary, we obtain a lower bound on the anchored isoperimetric dimension of the infinite component.