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Motivated by an evolutionary biology question, we study the followingproblem: we consider the hypercube $\{0,1\}^L$ where each node carries anindependent random variable uniformly distributed on $[0,1]$, except$(1,1,\ldots,1)$ which carries the value $1$ and $(0,0,\ldots,0)$ which carriesthe 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 valueson 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 independentexponential variables. As a first step in the analysis, we study the samequestion when the graph is that of a tree where the root has arity $L$, eachnode at level 1 has arity $L-1$, \ldots, and the nodes at level $L-1$ have onlyone offspring which are the leaves of the tree (all the leaves are assigned thevalue 1, the root the value $x\in[0,1]$).

The number of accessible paths in the hypercube