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.
A split tree of cardinality n is constructed by distributing n "balls" in a subset of vertices of an infinite tree which encompasses many types of random trees such as m-ary search trees, quad trees, median-of-(2k + 1) trees, fringe-balanced trees, digital search trees and random simplex trees. 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, the fluctuations of the size of the giant cluster as n → ∞ are described by an infinitely divisible distribution that belongs to the class of stable (asymmetric) Cauchy laws. This work generalizes the results for the random m-ary recursive trees by Berzunza (2015). 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 may be useful for studying percolation on other trees with logarithmic height; for instance in this work we study also the case of regular trees.
In this paper, we consider random trees associated with the genealogy of Crump-Mode-Jagers processes and perform Bernoulli bond-percolation whose parameter depends on the size of the tree. Our purpose is to show the existence of a giant percolation cluster for appropriate regimes as the size grows. We stress that the family trees of Crump-Mode-Jagers processes include random recursive trees, preferential attachment trees, binary search trees for which this question has been answered by Bertoin [7], as well as (more general) m-ary search trees, fragmentation trees, median-of-(2ℓ + 1) binary search trees, to name a few, where up to our knowledge percolation has not been studied yet.
The \(k\)-cut number of rooted graphs was introduced by Cai et al. as a generalization of the classical cutting model by Meir and Moon. In this paper, we show that all moments of the \(k\)-cut number of conditioned Galton-Watson trees converge after proper rescaling, which implies convergence in distribution to the same limit law regardless of the offspring distribution of the trees. This extends the result of Janson. Using the same method, we also show that the \(k\)-cut number of various random or deterministic trees of logarithmic height converges in probability to a constant after rescaling, such as random split-trees, uniform random recursive trees, and scale-free random trees.
The k-cut number of rooted graphs was introduced by Cai et al. [12] as a generalization of the classical cutting model by Meir and Moon [30]. In this paper, we show that all moments of the k-cut number of conditioned Galton-Watson trees converges after proper rescaling, which implies convergence in distribution to the same limit law regardless of the offspring distribution of the trees. This extends the result of Janson [25]. Using the same method, we also show that the k-cut number of various random or deterministic trees of logarithmic height converges in probability to a constant after rescaling, such as random split-trees, uniform random recursive trees, and scale-free random trees.
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