• Bernoulli
  • Volume 22, Number 2 (2016), 653-680.

The number of accessible paths in the hypercube

Julien Berestycki, Éric Brunet, and Zhan Shi

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Motivated by an evolutionary biology question, we study the following problem: we consider the hypercube $\{0,1\}^{L}$ where each node carries an independent random variable uniformly distributed on $[0,1]$, except $(1,1,\ldots,1)$ which carries the value $1$ and $(0,0,\ldots,0)$ which carries the value $x\in[0,1]$. We study the number $\Theta$ of paths from vertex $(0,0,\ldots,0)$ to the opposite vertex $(1,1,\ldots,1)$ along which the values on the nodes form an increasing sequence. We show that if the value on $(0,0,\ldots,0)$ is set to $x=X/L$ then $\Theta/L$ converges in law as $L\to\infty$ to $\mathrm{e}^{-X}$ times the product of two standard independent exponential variables.

As a first step in the analysis, we study the same question when the graph is that of a tree where the root has arity $L$, each node at level 1 has arity $L-1$, …, and the nodes at level $L-1$ have only one offspring which are the leaves of the tree (all the leaves are assigned the value 1, the root the value $x\in[0,1]$).

Article information

Bernoulli, Volume 22, Number 2 (2016), 653-680.

Received: November 2013
First available in Project Euclid: 9 November 2015

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branching processes evolutionary biology percolation trees


Berestycki, Julien; Brunet, Éric; Shi, Zhan. The number of accessible paths in the hypercube. Bernoulli 22 (2016), no. 2, 653--680. doi:10.3150/14-BEJ641.

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