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

• DOI: 10.48550/arxiv.2502.02535
• Publication Date: 2025-02-04
• Authors: Alexey Lotnikov; Anna Kotova
• Keywords: 60G50, 82B20, 82B27; Probability (math.PR); FOS: Mathematics; Mathematics - Probability

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