Algorithmic Pirogov-Sinai theory

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 Z d and on the torus (Z/nZ) 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/nZ) d at sufficiently low temperature. 1 n log Z T d n (λ) of the hard-core model on Z d . That is, we would like an algorithm which for any ǫ > 0 outputs a number η ∈ [f d (λ) − ǫ, f d (λ) + ǫ] and whose running time grows as slowly as possible as a function of 1/ǫ. Gamarnik and Katz [28] gave such an algorithm running in time polynomial in 1/ǫ for λ small enough that the strong spatial mixing holds; this condition implies the hard-core model is in the uniqueness regime. Adams, Briceño, Marcus, and Pavlov [1] gave a polynomial-time algorithm for approximating the free energy of the hard-core (and several other) models on Z 2 in a subset of the uniqueness regime. More interestingly, their results also apply for the hard-core and Widom-Rowlinson models on Z 2 in a subset of the non-uniqueness regime. The latter result is of a similar spirit to the results of this paper.

A Deterministic Algorithm for Counting Colorings with 2-Delta Colors

“…1 These algorithms have made use of one of two main techniques: decay of correlations, which exploits decreasing in uence of the spins (colors) on distant vertices on the spin at a given vertex; and polynomial interpolation, which uses the absence of zeros of the partition function in a suitable region of the complex plane. Early examples of the decay of correlations approach include [1,2,40], while for early examples of the polynomial interpolation method, we refer to the monograph of Barvinok [3] (see also, e.g., [4,13,25,27,30,34] for more recent examples). Unfortunately, however, in the case of colorings on general bounded degree graphs, these techniques have so far lagged well behind the MCMC algorithms mentioned above.…”

Counting independent sets in unbalanced bipartite graphs

“…On the other hand, previous algorithmic applications of the cluster expansion in low temperature (large λ) regimes [15,16] considered systems with multiple ground states, e.g. the all L or all R occupied independent sets for the hard-core model on a bipartite graph.…”

“…Following [15,16], our algorithms will be based on approximating the partition function of a polymer model [14,19] using the cluster expansion (for a textbook introduction to both polymer models and the cluster expansion see Chapter 5 of [10]).…”

“…(2) For each cluster Γ ∈ C m , compute φ(H(Γ)) and γ∈Γ w γ . It is shown in [17], using ideas and tools from [15,24,3], that this algorithm can be implemented with running time O n · (n/ǫ) O(log(∆ L ∆ R )/η) , which for ∆ L , ∆ R fixed is polynomial in n and 1/ǫ.…”

“…In most biregular cases our results give significant improvement to the parameters from [2], as we discuss below, and our algorithm does not require biregularity. More importantly, the method, based on the cluster expansion and the Kotecký-Preiss condition [19] and related to that of [15,16], gives detailed probabilistic information about the hard-core model in addition to the algorithmic results. In particular, we show that on bipartite graphs with parameters satisfying our conditions, the correlation between the occupancies of two vertices in the hard-core model decays exponentially fast in their distance.…”