Factor of iid percolation on trees

TL;DR

This paper investigates the maximum density of factor of iid percolation processes with finite clusters on large regular trees, establishing an asymptotic upper bound that matches known constructions.

Contribution

It provides an asymptotically tight upper bound on the density of factor of iid site percolation with finite clusters on large regular trees, linking it to independent sets.

Findings

Maximum density asymptotically at most (log d)/d as d grows

Bound is tight, achieved by independent sets

Implication for estimating maximal induced forests in regular graphs

Abstract

We study invariant percolation processes on the d-regular tree that are obtained as a factor of an iid process. We show that the density of any factor of iid site percolation process with finite clusters is asymptotically at most (log d)/d as d tends to infinity. This bound is asymptotically optimal as it can be realized by independent sets. One implication of the result is a (1/2)-factor approximation gap, asymptotically in d, for estimating the density of maximal induced forests in locally tree-like d-regular graphs via factor of iid processes.