The critical branching random walk in a random environment dies out

TL;DR

This paper proves that branching random walks in a random environment almost surely die out when the free energy parameter equals zero, resolving a conjecture and clarifying survival conditions.

Contribution

It establishes that, except for degenerate cases, BRWRE always die out at the critical free energy value, confirming a conjecture by Comets and Yoshida.

Findings

BRWRE die out when $ ext{ extPsi}=0$

Survival occurs only when $ ext{ extPsi}>0$

Complete proof of the conjecture by Comets and Yoshida

Abstract

We study the possibility for branching random walks in random environment (BRWRE) to survive. The particles perform simple symmetric random walks on the $d$-dimensional integer lattice, while at each time unit, they split into independent copies according to time-space i.i.d. offspring distributions. As noted by Comets and Yoshida, the BRWRE is naturally associated with the directed polymers in random environment (DPRE), for which the quantity $\Psi$ called the free energy is well studied. Comets and Yoshida proved that there is no survival when $\Psi<0$ and that survival is possible when $\Psi>0$. We proved here that, except for degenerate cases, the BRWRE always die when $\Psi=0$. This solves a conjecture of Comets and Yoshida.