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

Abstract

A split tree of cardinality n is constructed by distributing n "balls" (which often represent "key numbers") in a subset of vertices of an infinite tree. In this work, we study Bernoulli bond percolation on arbitrary split trees of large but finite cardinality n. We show for appropriate percolation regimes that depend on the cardinality n of the split tree that there exists a unique giant cluster that is of size comparable of that of the entire tree (where size is defined as either the number of vertices or the number of balls). The main result shows that in such percolation regimes, also known as supercritical regimes, the fluctuations of the size of the giant cluster are non-Gaussian as n → ∞. 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. Our approach is based on a remarkable decomposition of the size of the giant percolation cluster as a sum of essentially independent random variables which allows us to apply a classical limit theorem for the convergence of triangular arrays to infinitely divisible distributions. This may be of independent interest and it may be useful for studying percolation on other classes of trees with logarithmic height, for instance in this work we study also the case of regular trees.

The asymptotic distribution of cluster sizes for supercritical percolation on random split trees

“…1,n → e −c/µ , in distribution, as n → ∞, which proves the second assertion. Moreover,[6, Lemma 2] also shows that C * 1 /n 1,n → 0, in distribution, as n → ∞, and by Corollary 1, we conclude that C * 1 = o p (n/ ln n). This completes the proof.d − → (x 1 , .…”

Tree limits and limits of random trees

“…Without going into details, we note that the proof of (13.5) in [7], as well as our similar proof in Section 13.1, is based on showing the two results (12.3) and (12.4), and that (12.3) was shown by Holmgren [23].…”

The asymptotic distribution of cluster sizes for supercritical percolation on random split trees

“…Proof of Theorem 2. The first claim has been proved in [6,Lemma 3], and thus, we only prove the second one. Following exactly the same argument as in the proof of Corollary 2, we deduce from Corollary 1 and Proposition 3 that for every fixed i ∈ N,…”

“…That is, [6, Lemma 3] implies that C j ∕n j,n P → 𝛼e −c∕𝜇 , as n → ∞ which proves the second assertion. Moreover, [6,Lemma 3] also shows that C * j ∕n j,n P → 0, as n → ∞, and by Corollary 1, C * j = o p (n∕ ln n). ▪…”