laboratoire de physique statistique
 
 
laboratoire de physique statistique

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BERNOULLI 


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2016
The number of accessible paths in the hypercube - Berestycki, Julien and Brunet, Eric and Shi, Zhan
BERNOULLI 22653-680 (2016)

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,..., 1) which carries the value 1 and (0, 0,..., 0) which carries the value x is an element of [0, 1]. We study the number Theta 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 Theta/L converges in law as L -> infinity 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,..., 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 is an element of [0, 11).