Localization for Linear Stochastic Evolutions

TL;DR

This paper establishes a link between slow population growth and localization in a stochastic lattice growth model, even when extinction occurs with positive probability, providing new insights into the behavior of such systems.

Contribution

It introduces a novel equivalence between population growth rate and localization, applicable to models with finite-time extinction probability.

Findings

Proves the equivalence between slow growth and localization via replica overlap.

Characterizes the event where an exponential martingale vanishes.

Extends the understanding of stochastic growth models with extinction.

Abstract

We consider a discrete-time stochastic growth model on the $d$-dimensional lattice with non-negative real numbers as possible values per site. The growth model describes various interesting examples such as oriented site/bond percolation, directed polymers in random environment, time discretizations of the binary contact path process. We show the equivalence between the slow population growth and a localization property in terms of "replica overlap". The main novelty of this paper is that we obtain this equivalence even for models with positive probability of extinction at finite time. In the course of the proof, we characterize, in a general setting, the event on which an exponential martingale vanishes in the limit.