A stochastic combustion model with thresholds on trees

TL;DR

This paper introduces a stochastic combustion model on trees where particles activate upon reaching visit thresholds, revealing a phase transition in total root visits on infinite d-ary trees.

Contribution

It models a new threshold-based activation process on trees and demonstrates a phase transition phenomenon in the total root visits.

Findings

Total root visits exhibit a phase transition.

The model applies to infinite d-ary trees.

Activation depends on visit thresholds.

Abstract

Place one active particle at the root of a graph and a Poisson-distributed number of dormant particles at the other vertices. Active particles perform simple random walk. Once the number of visits to a site reaches a random threshold, any dormant particles there become active. For this process on infinite $d$-ary trees, we show that total root visits undergoes a phase transition.