The sustainability probability for the critical Derrida-Retaux model

Abstract

We are interested in the recursive model $(Y_n,n\ge 0)$ studied by Collet, Eckmann, Glaser and Martin [9] and by Derrida and Retaux [12]. We prove that at criticality, the probability $\mathbf{P}(Y_n>0)$ behaves like $n^{-2+o(1)}$ as $n$ goes to infinity; this gives a weaker confirmation of predictions made in [9], [12] and [6]. Our method relies on studying the number of pivotal vertices and open paths, combined with a delicate coupling argument.

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