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,…,1) which carries the value 1 and (0,0,…,0) which carries
the value x∈[0,1]. We study the number Θ of paths from vertex
(0,0,…,0) to the opposite vertex (1,1,…,1) along which the values
on the nodes form an increasing sequence. We show that if the value on
(0,0,…,0) is set to x=X/L then Θ/L converges in law as
L→∞ to 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∈[0,1]).Comment: Published at http://dx.doi.org/10.3150/14-BEJ641 in the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm