Can the root cluster remain largest forever in random recursive tree percolation?

Abstract

We consider Bernoulli bond percolation with fixed retention parameter $p$ on the random recursive tree, coupled through the natural growth process. We prove that the root cluster has a strictly positive probability of remaining a largest cluster at every time. Equivalently, in the associated Simon-type Chinese restaurant process, the first table has a positive probability of remaining a largest table forever. We further show that two associated quantities, the limit as $n\to\infty$ of the probability that the root cluster is a largest cluster at time $n$ and the probability that it remains a largest cluster for all times, are strictly increasing and continuous functions of $p$ on $(0,1]$, and both tend to zero as $p\downarrow0$.