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Comparison of partition functions in a space–time random environment

Abstract

Let \(Z^1\) and \(Z^2\) be partition functions in the random polymer model in the same environment but driven by different underlying random walks. We give a comparison in concave stochastic order between \(Z^1\) and \(Z^2\) if one of the random walks has “more randomness” than the other. We also treat some related models: The parabolic Anderson model with space–time Lévy noise; Brownian motion among space–time obstacles; and branching random walks in space–time random environments. We also obtain a necessary and sufficient criterion for \(Z^1\preceq _{cv}Z^2\) if the lattice is replaced by a regular tree.

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Acknowledgements

The author would like to thank Noam Berger, David Criens, Lexuri Fernandez and Nina Gantert for carefully proofreading the manuscript and for many helpful comments.

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Correspondence to Stefan Junk.

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Communicated by Eric A. Carlen.

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Junk, S. Comparison of partition functions in a space–time random environment. J Stat Phys 181, 95–115 (2020). https://doi.org/10.1007/s10955-020-02566-4

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  • DOI: https://doi.org/10.1007/s10955-020-02566-4

Keywords

  • Directed polymers
  • Random environment
  • Stochastic order

Mathematics Subject Classification

  • 60K37
  • 60E15