Increasing paths in regular trees
Lee Zhuo Zhao (University of Cambridge)
Abstract
We consider a regular $n$-ary tree of height $h$, for which every vertex except the root is labelled with an independent and identically distributed continuous random variable. Taking motivation from a question in evolutionary biology, we consider the number of paths from the root to a leaf along vertices with increasing labels. We show that if $\alpha = n/h$ is fixed and $\alpha > 1/e$, the probability that there exists such a path converges to $1$ as $h \to \infty$. This complements a previously known result that the probability converges to $0$ if $\alpha \leq 1/e$.
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Pages: 1-10
Publication Date: November 9, 2013
DOI: 10.1214/ECP.v18-2784
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