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The Toom Interface via Coupling

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Abstract

We consider a one dimensional interacting particle system which describes the effective interface dynamics of the two dimensional Toom model at low noise. We prove a number of basic properties of this model. First we consider the dynamics on a finite interval [1, N) and bound the mixing time from above by 2N. Then we consider the model defined on the integers. Because the interaction range of the rates and the jump sizes can be arbitrarily large, this is a non-Feller process. As such, we can define the process starting from product Bernoulli measures with density \(p \in (0, 1)\), but not from arbitrary measures. We show that the only possible invariant measures are those product Bernoulli measures, under a modest technical condition. We further show that the unique stationary measure on \([0, \infty )\) converges to a product Bernoulli measure with fixed density when viewed far from 0.

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Notes

  1. Pronounce Toom with a long o, not with the English pronunciation of oo

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Acknowledgements

We thank Joel Lebowitz for motivating us to work on the problem and for telling us that the i.i.d. measure is invariant on the whole line. We thank Christian Maes for pointing out the relation with reference [2]. A special thanks goes to Wojciech de Roeck, who was a coauthor on previous versions of this paper, before electing to resign. NC is supported by Israel Science Foundation Grant Nos. 915/12 and 1692/17. GK is supported by Israel Science Foundation Grant No. 1369/15 and by the Jesselson Foundation.

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Authors and Affiliations

  1. Department of Mathematics, The Technion, Haifa, Israel

    Nicholas Crawford

  2. Department of Math and CS, The Weizmann Institute, Rehovot, Israel

    Gady Kozma

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  1. Nicholas Crawford
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  2. Gady Kozma
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Correspondence to Nicholas Crawford.

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Communicated by Pablo A Ferrari.

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Crawford, N., Kozma, G. The Toom Interface via Coupling. J Stat Phys 179, 408–447 (2020). https://doi.org/10.1007/s10955-020-02529-9

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  • DOI: https://doi.org/10.1007/s10955-020-02529-9

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