Frog model on $\mathbb{Z}$ with random survival parameter

We study the frog model on $\mathbb{Z}$ with geometric lifetimes, introducing a random survival parameter. Active and inactive particles are placed at the vertices of $\mathbb{Z}$. The lifetime of each active particle follows a geometric random variable with parameter $1-p$, where $p$ is randomly sampled from a distribution $\pi$. Each active particle performs a simple random walk on $\mathbb{Z}$ until it dies, activating any inactive particles it encounters along its path. In contrast to the usual case where $p$ is fixed, we show that there exist non-trivial distributions $\pi$ for which the model survives with positive probability. More specifically, for $\pi \sim Beta(\alpha ,\beta )$, we establish the existence of a critical value $\beta =0.5$, that separates almost sure extinction from survival with positive probability. Furthermore, we show that the model is recurrent whenever it survives with positive probability.