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

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