Metastable behavior for bootstrap percolation on regular trees

TL;DR

This paper studies the metastable transition in bootstrap percolation on regular trees, showing how the system lingers near a critical state before rapidly becoming fully occupied as the initial occupation probability approaches a threshold.

Contribution

It provides a detailed analysis of the metastable behavior and transition times in bootstrap percolation on regular trees near the critical point.

Findings

System remains near the critical state for time proportional to h^{-1/2} as p approaches p_t from above.

Transition to full occupation occurs in constant time O(1).

The configuration law converges to a well-defined measure as h tends to zero.

Abstract

We examine bootstrap percolation on a regular (b+1)-ary tree with initial law given by Bernoulli(p). The sites are updated according to the usual rule: a vacant site becomes occupied if it has at least theta occupied neighbors, occupied sites remain occupied forever. It is known that, when b>theta>1, the limiting density q=q(p) of occupied sites exhibits a jump at some p_t=p_t(b,theta) in (0,1) from q_t:=q(p_t)<1 to q(p)=1 when p>p_t. We investigate the metastable behavior associated with this transition. Explicitly, we pick p=p_t+h with h>0 and show that, as h decreases to 0, the system lingers around the "critical" state for time order h^{-1/2} and then passes to fully occupied state in time O(1). The law of the entire configuration observed when the occupation density is q in (q_t,1) converges, as h tends to 0, to a well-defined measure.