The sustainability probability for the critical Derrida-Retaux model

TL;DR

This paper analyzes the critical behavior of a recursive Derrida-Retaux model, proving that the probability of positivity decays polynomially with an exponent close to -2, confirming some theoretical predictions.

Contribution

It establishes the asymptotic decay rate of the survival probability at criticality for the Derrida-Retaux model using novel coupling and pivotal vertex analysis.

Findings

Probability ${f P}(Y_n>0)$ behaves like $n^{-2+o(1)}$ at criticality

Provides a weaker confirmation of earlier predictions about the model's behavior

Introduces a method combining pivotal vertices and coupling arguments

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 ${\bf P}(Y_n>0)$ 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.