Poisson approximation for large clusters in the supercritical FK model

TL;DR

This paper demonstrates that in the supercritical FK model, the distribution of large finite clusters can be approximated by a Poisson process under certain mixing conditions, using the Chen-Stein method.

Contribution

It introduces a Poisson approximation for large clusters in the supercritical FK model leveraging the Chen-Stein method and weak mixing properties.

Findings

Large finite clusters follow a Poisson distribution under weak mixing.

The Chen-Stein method effectively approximates cluster distributions.

The approach applies to the supercritical FK model with specific mixing conditions.

Abstract

Using the Chen-Stein method, we show that the spatial distribution of large finite clusters in the supercritical FK model approximates a Poisson process when the ratio weak mixing property holds.