## Electronic Communications in Probability

### Increasing paths in regular trees

#### 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 $\alpha \leq 1/e$.

#### Article information

Source
Electron. Commun. Probab., Volume 18 (2013), paper no. 87, 10 pp.

Dates
Accepted: 9 November 2013
First available in Project Euclid: 7 June 2016

https://projecteuclid.org/euclid.ecp/1465315626

Digital Object Identifier
doi:10.1214/ECP.v18-2784

Mathematical Reviews number (MathSciNet)
MR3141796

Zentralblatt MATH identifier
1306.60128

Rights

#### Citation

Roberts, Matthew; Zhao, Lee. Increasing paths in regular trees. Electron. Commun. Probab. 18 (2013), paper no. 87, 10 pp. doi:10.1214/ECP.v18-2784. https://projecteuclid.org/euclid.ecp/1465315626

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