The number of accessible paths in the hypercube

TL;DR

This paper investigates the asymptotic distribution of the number of increasing paths in a hypercube with random node values, revealing a limiting distribution involving exponential variables as dimension grows.

Contribution

It introduces a novel probabilistic analysis of accessible paths in high-dimensional hypercubes, connecting the problem to exponential distributions and providing new asymptotic results.

Findings

The scaled number of accessible paths converges in distribution to an exponential-related random variable.

The initial node value scaled by dimension influences the limiting distribution.

The analysis extends to a tree structure, providing insights into the hypercube problem.

Abstract

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 $L-1$, \ldots, 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 $x\in[0,1]$).