Infinite collisions of simple random walks on random recursive trees generated by Bernoulli sequences

Abstract

In this paper, we study random recursive trees generated by Bernoulli sequences. Starting from a graph with two vertices and one edge, each new vertex is connected to the last vertex with probability $p$, or to the second-last vertex with probability $q=1-p$, this recursive construction yields a random infinite recursive tree $T$. We prove that $T$ almost surely has exactly one topological end. Furthermore, we establish that $T$ has the infinite collision property: two independent simple random walks on $T$ collide infinitely often almost surely.

13pages