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

TL;DR

This work analyzes the Derrida-Retaux model with a random number of terms by linking the free energy $Q=\lim_{n\to\infty}\frac{\mathbb{E}(X_n)}{(\mathbb{E}N)^n}$ to the initial data via moment generating functions. The authors introduce a key functional $D_0(s,m)=(m-1)sF'_0(s)-aF_0(s)$ and derive two sufficient conditions: (i) $D_0(\mathbb{E}N^{1/a},\mathbb{E}N)>0$ implies a supercritical regime ($Q>0$), and (ii) if $N\le M$ a.s. and $D_0(1+(M-1)/a,M)<0$, then the system is subcritical ($Q=0$). These results hold in both random-$N$ and fixed-$N$ settings, with the fixed-$N$ case giving $D_0(s)=(n-1)sF'_0(s)-aF_0(s)$. The proofs rely on recursive evolution equations for the moment generating functions and monotone bounds on the derived functionals $D_n(s)$, connecting local generating-function dynamics to global phase behavior, and generalize known Lifshits-type criteria to arbitrary $a\ge1$ and random term counts.

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.

Theorems & Definitions (6)

• Theorem 1
• Theorem 2
• Lemma 1
• Lemma 2
• Lemma 3
• Lemma 4