Localization for a Class of Linear Systems

TL;DR

This paper investigates the relationship between population growth and localization in certain stochastic lattice models, establishing equivalences and stronger localization properties under specific conditions.

Contribution

It demonstrates the equivalence between slow growth and localization, and proves a stronger form of localization for a class of stochastic growth models.

Findings

Slow population growth is equivalent to localization.

The spatial distribution remains non-uniform over time.

Under certain assumptions, stronger localization properties are established.

Abstract

We consider a class of continuous-time stochastic growth models on $d$-dimensional lattice with non-negative real numbers as possible values per site. The class contains examples such as binary contact path process and potlatch process. We show the equivalence between the slow population growth and localization property that the time integral of the replica overlap diverges. We also prove, under reasonable assumptions, a localization property in a stronger form that the spatial distribution of the population does not decay uniformly in space.