Fires on large recursive trees

TL;DR

This paper studies the spread of fires on large recursive trees, revealing phase transitions, connectivity properties of fireproof subtrees, and the size distribution of burnt areas as the tree size grows.

Contribution

It introduces a probabilistic model of fire spread on recursive trees, analyzing phase transitions and structural properties of fireproof and burnt components.

Findings

Phase transition at $p_n$ of order $rac{ ext{ln} n}{n}$

Existence of a giant fireproof component under certain conditions

Distribution of burnt subtree sizes

Abstract

We consider random dynamics on a uniform random recursive tree with $n$ vertices. Successively, in a uniform random order, each edge is either set on fire with some probability $p_n$ or fireproof with probability $1-p_n$. Fires propagate in the tree and are only stopped by fireproof edges. We first consider the proportion of burnt and fireproof vertices as $n\to\infty$, and prove a phase transition when $p_n$ is of order $\ln n/n$. We then study the connectivity of the fireproof forest, more precisely the existence of a giant component. We finally investigate the sizes of the burnt subtrees.