Criticality conditions in the Derrida-Retaux model with a random number of terms

TL;DR

This paper investigates the Derrida-Retaux model with a random number of terms, establishing conditions for subcritical and supercritical regimes based on initial distributions and the randomness of the number of terms.

Contribution

It provides new sufficient conditions for the criticality regimes in the Derrida-Retaux model with a stochastic number of terms, extending previous deterministic analyses.

Findings

Identifies conditions for subcritical regime ($Q=0$).

Identifies conditions for supercritical regime ($Q>0$).

Defines the energy limit in the model.

Abstract

The article considers the Derrida-Retaux model with a random number of terms, i.e. a sequence of integer random variables defined by the relations $ X_{n + 1} = (X_n^{(1)} + X_n^{(2)} + ... + X_n^{(N_n)} - a)^{+}$, $n\ge 0$, where $X_n^{j}$ are independent copies of $X_n$, the values of $N_j$ are independent and identically distributed, $a$ is a positive integer. The energy in the model is defined as $Q:=\lim\limits_{n\to\infty} \frac{\mathbb{E}(X_{n})}{(\mathbb{E}N_1)^{n}}$. We present sufficient conditions (in terms of distributions of $X_0$ and $N_1$) for subcritical ($Q=0$) and supercritical ($Q>0$) regimes of model behavior.