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 et al. (Commun Math Phys 94:353–370, 1984) and by Derrida and Retaux (J Stat Phys 156:268–290, 2014). 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 Collet et al. (1984), Derrida and Retaux (2014) and Chen et al. (in: Sidoravicius (ed) Sojourns in probability theory and statistical physics-III, Springer, Singapore, 2019). Our method relies on studying the number of pivotal vertices and open paths, combined with a delicate coupling argument.