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Algorithmic Pirogov–Sinai theory

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Abstract

We develop an efficient algorithmic approach for approximate counting and sampling in the low-temperature regime of a broad class of statistical physics models on finite subsets of the lattice \(\mathbb {Z}^d\) and on the torus \((\mathbb {Z}/n\mathbb {Z})^d\). Our approach is based on combining contour representations from Pirogov–Sinai theory with Barvinok’s approach to approximate counting using truncated Taylor series. Some consequences of our main results include an FPTAS for approximating the partition function of the hard-core model at sufficiently high fugacity on subsets of \(\mathbb {Z}^d\) with appropriate boundary conditions and an efficient sampling algorithm for the ferromagnetic Potts model on the discrete torus \((\mathbb {Z}/n\mathbb {Z})^d\) at sufficiently low temperature.

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Notes

  1. This means \(\frac{|\partial ^{\mathrm{in}}\varLambda _{n}|}{|\varLambda _{n}|}\rightarrow 0\), see [23, Section 3.2.1].

  2. We could consider only vertices x such that \(d_\infty (x, \varLambda ^c)>1\), but it does no harm to include the others.

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Acknowledgements

WP and GR thank Ivona Bezáková, Leslie Goldberg, and Mark Jerrum for organizing the 2017 Dagstuhl seminar on computational counting and Jan Hladkỳ for organizing the 2018 workshop on graph limits in Bohemian Switzerland. Both meetings provided essential inspiration and discussion leading to this work. TH thanks Roman Kotecký for helpful discussions. We thank Eric Vigoda, Matthew Jenssen, and Reza Gheissari for detailed comments on a draft of the paper. We are moreover grateful to the anonymous referees for their helpful suggestions.

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  1. University of Bristol, Bristol, UK

    Tyler Helmuth

  2. University of Illinois at Chicago, Chicago, USA

    Will Perkins

  3. University of Amsterdam, Amsterdam, The Netherlands

    Guus Regts

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  1. Tyler Helmuth
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Tyler Helmuth: Supported by EPSRC Grant EP/P003656/1.

Will Perkins: Supported in part by EPSRC Grant EP/P009913/1 and NSF Career Award DMS-1847451.

Guus Regts: Supported by an NWO Veni Grant.

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Helmuth, T., Perkins, W. & Regts, G. Algorithmic Pirogov–Sinai theory. Probab. Theory Relat. Fields 176, 851–895 (2020). https://doi.org/10.1007/s00440-019-00928-y

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