@article{ECP2784,
author = {Matthew Roberts and Lee Zhao},
title = {Increasing paths in regular trees},
journal = {Electron. Commun. Probab.},
fjournal = {Electronic Communications in Probability},
volume = {18},
year = {2013},
keywords = {evolutionary biology; trees; branching processes; increasing paths},
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 $\alpha \leq 1/e$.},
pages = {no. 87, 1-10},
issn = {1083-589X},
doi = {10.1214/ECP.v18-2784},
url = {http://ecp.ejpecp.org/article/view/2784}}