Critical branching processes in random environment and Cauchy domain of attraction

We are interested in the survival probability of a population modeled by a critical branching process in an i.i.d. random environment. We assume that the random walk associated with the branching process is oscillating and satisfies a Spitzer condition $\mathbf{P}(S_{n}>0)\rightarrow \rho ,\ n\rightarrow \infty$, which is a standard condition in fluctuation theory of random walks. Unlike the previously studied case $\rho \in (0,1)$, we investigate the case where the offspring distribution is in the domain of attraction of a stable law with parameter $1$, which implies that $\rho =0$ or $1$. We find the asymptotic behaviour of the survival probability of the population in these two cases.