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 $X_{n+1}=(X_n^{(1)}+X_n^{(2)}+\mathrm{.}\mathrm{.}\mathrm{.}+X_n^{(N_n)}-a{)}^{+}$, $n\ge 0$, 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 $Q:=\underset{n\to \mathrm{\infty }}{\lim }\frac{\mathbb{E}(X_n)}{(\mathbb{E}{N}_1{)}^n}$. 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.