Fires on large recursive trees

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.

Accepted for publication in Stochastic Processes and their Applications. 24 pages, 4 figures

References in corpus (1)

• Multiple isolation of nodes in recursive trees

Cited by in corpus (3)

• Random Recursive Trees and the Elephant Random Walk
• Asymptotic results on Hoppe trees and its variations
• Some remarks on biased recursive trees