Algorithmic Pirogov-Sinai theory

06/29/2018

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by   Tyler Helmuth, et al.

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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 Z^d and on the torus ( Z/n 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 Z^d with appropriate boundary conditions and an efficient sampling algorithm for the ferromagnetic Potts model on the discrete torus ( Z/n Z)^d at sufficiently low temperature.