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Localization for a class of linear systems. (English) Zbl 1226.60134

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

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60J25 Continuous-time Markov processes on general state spaces
60J75 Jump processes (MSC2010)
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