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 pn or fireproof with probability 1−pn. Fires propagate in the tree and are only stopped by fireproof edges. We first consider the proportion of burnt and fireproof vertices as n→∞, and prove a phase transition when pn is of order lnn/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.