Algorithmic Pirogov–Sinai theory

  • Tyler HelmuthEmail author
  • Will Perkins
  • Guus Regts


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.


Approximate sampling Approximation algorithms FPTAS Discrete spin systems Pirogov–Sinai theory Cluster expansion 

Mathematics Subject Classification

82B20 68W25 60K35 



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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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.University of BristolBristolUK
  2. 2.University of Illinois at ChicagoChicagoUSA
  3. 3.University of AmsterdamAmsterdamThe Netherlands

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