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

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)} +\cdots + 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 $Q:=\lim\limits_{n\to\infty} \frac{\mathbf{E} (X_{n})}{(\mathbf{E} N_1)^{n}}$. 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.