Yule processes with rare mutation and their applications to percolation on $b$-ary trees

Abstract

We consider supercritical Bernoulli bond percolation on a large b-ary tree, in the sense that with high probability, there exists a giant cluster. We show that the size of the giant cluster has non-gaussian fluctuations, which extends a result due to Schweinsberg [15] in the case of random recursive trees. Using ideas in the recent work of Bertoin and Uribe Bravo [5], the approach developed in this work relies on the analysis of the sub-population with ancestral type in a system of branching processes with rare mutations, which may be of independent interest. This also allows us to establish the analogous result for scale-free trees.

The fluctuations of the giant cluster for percolation on random split trees

“…Instead, they are described by an infinitely divisible distribution that belongs to the class of stable Cauchy laws. This work is a generalization of the results for the random m-ary recursive trees in Berzunza [6], which is one specific case of split trees. Other important examples of split trees include m-ary search trees, quad trees, median-of-(2k + 1) trees, fringe-balanced trees, digital search trees and random simplex trees.…”

The existence of a giant cluster for percolation on large Crump–Mode–Jagers trees

“…Thus we have to use different tools, although some guidelines are similar to [10]. We stress that similar connections with systems of (Markovian) branching processes have been used before to study percolation on random recursive trees [4,5] and m-ary random increasing trees [11].…”

“…By Theorem 1, we only need to computeμ(1) and E[φ (1)]. Note that (8) and(11) show thatμ(1) = −μ (1) = H 2 +2 − H +1 . Note also that…”

Weak limits for the largest subpopulations in Yule processes with high mutation probabilities

“…The strategy we develop here in terms of Yule processes allows a concise analysis of cluster sizes, for any choice of p n tending to zero. For sequences of p n such that p n → 1 or p n = p ∈ (0, 1) remains constant, similar connections between systems of branching processes and percolation on increasing tree families have been utilized before in, e.g., [9,8,4,5]. The precise definition of a random recursive tree, its connection to Yule processes and more references to existing results on percolation will be discussed in Section 5.…”

“…We stress that there are other natural families of trees which can be grown according to a probabilistic evolution algorithm, like, e.g., scale-free random trees or b-ary random increasing trees. Indeed, branching systems with mutations were used in [8] to study percolation on scale-free random trees when p n ∼ a/ ln n, and in a similar fashion by Berzunza in [9] for bary random increasing trees. In the case of random recursive trees, the underlying population system is particularly simple, so we restricted our discussion of the subcritical regime to these trees, but we certainly expect similar results to hold true for other classes of increasing tree families.…”