The critical branching random walk in a random environment dies out

Olivier Garet,Régine Marchand

doi:10.1214/ecp.v18-2438

Olivier Garet, Régine Marchand

Open Access

https://doi.org/10.1214/ecp.v18-2438

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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 $\Psi 0$. We proved here that, except for degenerate cases, the BRWRE always die when $\Psi=0$. This solves a conjecture of Comets and Yoshida.

Highlights

Comets–Yoshida Theorem [3] relates the survival of the branching random walks in random environment (BRWRE) with a functional on an associated directed polymer in random environment as follows: define on a probability space (ΩS, FS, PS) a simple symmetric random walk (St)t≥0 on Zd starting from S0 = 0
It is easy to see (e.g., [9, Lemma 1.4]) that Zt is the expectation of the number of particles of the BRWRE living at time t, knowing the random environment q =(t,x)∈N×Zd
We encode the BRWRE starting from the initial configuration A by the random variables ηtA = (ηtA(x))t∈N,x∈Zd which represent the number of particles living on site x at time t

Summary

Introduction

Comets–Yoshida Theorem [3] relates the survival of the BRWRE with a functional on an associated directed polymer in random environment as follows: define on a probability space (ΩS , FS, PS) a simple symmetric random walk (St)t≥0 on Zd starting from S0 = 0. It is easy to see (e.g., [9, Lemma 1.4]) that Zt is the expectation of the number of particles of the BRWRE living at time t, knowing the random environment q = (qt,x)(t,x)∈N×Zd. Note that (2), combined with the inequality | log u| ≤ u ∨ u−1 for u > 0, implies that. We encode the BRWRE starting from the initial configuration A by the random variables ηtA = (ηtA(x))t∈N,x∈Zd which represent the number of particles living on site x at time t.

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