A local limit theorem for triple connections in subcritical Bernoulli percolation

TL;DR

This paper establishes a local limit theorem describing the asymptotic probability that a site is connected to three points via disjoint paths in subcritical Bernoulli percolation on high-dimensional integer lattices.

Contribution

It introduces a new local limit theorem for triple connections in subcritical Bernoulli percolation, extending understanding of connection probabilities in large-scale regimes.

Findings

Proves a local limit theorem for triple connections

Analyzes behavior in subcritical Bernoulli percolation on $\\mathbb{Z}^{d}$

Focuses on the limit where distances between points tend to infinity

Abstract

We prove a local limit theorem for the probability of a site to be connected by disjoint paths to three points in subcritical Bernoulli percolation on $\mathbb{Z}^{d},\,d\geq2$ in the limit where their distances tend to infinity.