Asymptotic behavior of the generalized Derrida-Retaux recursive model

TL;DR

This paper investigates the long-term asymptotic behavior of a generalized recursive model related to Derrida-Retaux, providing detailed expansions and analyzing key probabilistic properties.

Contribution

It introduces asymptotic expansions for the model's parameters, extending understanding of the model's long-term behavior and its distributional properties.

Findings

Asymptotic expansions of model parameters are derived.

The sustainability probability's asymptotics are characterized.

The marginal distribution and generating function behaviors are analyzed.

Abstract

We study the max-type recursive model introduced by Hu and Shi (J. Stat. Phys., 2018), which generalizes the model of Derrida and Retaux (J. Stat. Phys., 2014). The class of geometric-type marginal distributions is preserved by the model with a geometric offspring distribution. We give some long-time asymptotic expansions of the parameters of the marginal distribution. From the expansions, we derive the asymptotics of the sustainability probability, marginal distribution, first moment and probability generating function.