Finite Size Percolation in Regular Trees

TL;DR

This paper investigates the behavior of the largest cluster size and longest run in percolation on regular trees, providing convergence results as the number of generations increases.

Contribution

It offers new theoretical insights into the asymptotic properties of percolation clusters in regular trees.

Findings

Largest cluster size converges almost surely

Longest run length converges weakly

Asymptotic behavior characterized for large generations

Abstract

In the context of percolation in a regular tree, we study the size of the largest cluster and the length of the longest run starting within the first d generations. As d tends to infinity, we prove almost sure and weak convergence results.