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A split tree of cardinality $n$ is constructed by distributing $n$ "balls" ina subset of vertices of an infinite tree which encompasses many types of randomtrees such as $m$-ary search trees, quad trees, median-of-$(2k+1)$ trees,fringe-balanced trees, digital search trees and random simplex trees. In thiswork, we study Bernoulli bond percolation on arbitrary split trees of large butfinite cardinality $n$. We show for appropriate percolation regimes that dependon the cardinality $n$ of the split tree that there exists a unique giantcluster, the fluctuations of the size of the giant cluster as $n \rightarrow\infty$ are described by an infinitely divisible distribution that belongs tothe class of stable Cauchy laws. This work generalizes the results for therandom $m$-ary recursive trees in Berzunza (2015). Our approach is based on aremarkable decomposition of the size of the giant percolation cluster as a sumof essentially independent random variables which may be useful for studyingpercolation on other trees with logarithmic height; for instance in this workwe study also the case of regular trees.

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