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Phase Transition in Dependent Percolation

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Abstract

In this paper we discuss two different models of dependent percolation on the graph ℤ2. These models can be thought of as percolation in a random environment. They were inspired by the work of McCoy and Wu [7,8] on the Ising model in a random environment as well as other models of particle systems in a random environment [9, 5, 6, 3]. We show that both models of dependent percolation exhibit phase transitions. This proves a version of stability for percolation on ℤ2 and proves a conjecture of Jonasson, Mossel and Peres [4], who proved a similar result on ℤ3.

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Authors and Affiliations

  1. Department of Mathematics, University of Washington, Seattle, WA, 98195, USA

    Christopher Hoffman

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  1. Christopher Hoffman
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Correspondence to Christopher Hoffman.

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Communicated by M. Aizenman

Research supported in part by an NSF postdoctoral fellowship

Acknowledgement I would like to thank David Levin and Yuval Peres for introducing me to the problem. I would also like to thank Yuval Peres and Eric Babson for helpful conversations.

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Hoffman, C. Phase Transition in Dependent Percolation. Commun. Math. Phys. 254, 1–22 (2005). https://doi.org/10.1007/s00220-004-1240-2

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  • DOI: https://doi.org/10.1007/s00220-004-1240-2

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