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Localization for Linear Stochastic Evolutions

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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.

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Authors and Affiliations

  1. Division of Mathematics, Graduate School of Science, Kyoto University, Kyoto, 606-8502, Japan

    Nobuo Yoshida

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  1. Nobuo Yoshida
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Correspondence to Nobuo Yoshida.

Additional information

The author was supported in part by JSPS Grant-in-Aid for Scientific Research, Kiban (C) 17540112.

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Yoshida, N. Localization for Linear Stochastic Evolutions. J Stat Phys 138, 598–618 (2010). https://doi.org/10.1007/s10955-009-9876-0

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  • DOI: https://doi.org/10.1007/s10955-009-9876-0

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