Counting without sampling: Asymptotics of the log‐partition function for certain statistical physics models

Abstract

ABSTRACT:In this article we propose new methods for computing the asymptotic value for the logarithm of the partition function (free energy) for certain statistical physics models on certain type of finite graphs, as the size of the underlying graph goes to infinity. The two models considered are the hard-core (independent set) model when the activity parameter λ is small, and also the Potts (q-coloring) model. We only consider the graphs with large girth. In particular, we prove that asymptotically the logarithm of the number of independent sets of any r-regular graph with large girth when rescaled is approximately constant if r ≤ 5. For example, we show that every 4-regular n-node graph with large girth has approximately (1.494 · · · ) n -many independent sets, for large n. Further, we prove that for every r-regular graph with r ≥ 2, with n nodes and large girth, the number of proper q ≥ r + 1 colorings is approximatelyn , for large n. We also show that these results hold for random regular graphs with high probability (w.h.p.) as well.As a byproduct of our method we obtain simple algorithms for the problem of computing approximately the logarithm of the number of independent sets and proper colorings, in low degree COUNTING WITHOUT SAMPLING 453graphs with large girth. These algorithms are deterministic and use certain correlation decay properties for the corresponding Gibbs measures, and its implications to uniqueness of the Gibbs measures on the infinite trees, as well as some simple cavity trick which is well known in the physics and the Markov chain sampling literature.

Spatial mixing and the connective constant: optimal bounds

“…this line of work culminated in papers by Dyer and Greenhill [5] and Vigoda [29], who gave MCMC based FPRASs for λ < 2/(d − 1) for graphs of maximum degree at most d + 1. Weitz [31] (see also [2]) introduced a new paradigm by using correlation decay directly to design a deterministic FPTAS and gave an algorithm under the condition λ < λ c (d) for graphs of degree at most d + 1; this range of applicability was later proved to be optimal by Sly [26] (see also [7,27]). To date, no MCMC based algorithm is known to have a range of applicability as wide as Weitz's algorithm.…”

Extremal Regular Graphs: Independent Sets and Graph Homomorphisms

“…The following inequality was proved by Csikvári and Lin [15]. For q ≥ d + 1, the constant in the inequality is best possible as it is the limit for any sequence of d-regular graphs with increasing girth [3]. Theorem 8.4 ([15]).…”

Replica symmetry of the minimum matching

“…Within the framework of the replica method, Mézard and Parisi obtained an analytical characterization of β(d) which is conjectured to be correct for all d > 0. They arrived at an integral equation equivalent to (7) F (x) = exp…”

“…However, the proofs in [20], [28], [41], [42] are very different from the approach in the physics literature and do not seem to generalize to d = 1. The original proof by Aldous [2] is the one that comes closest to justifying the replica symmetric ansatz (particularly in view of additional results in [6], [37]), but it seems to rely on finding a solution to (7).…”

“…These "recursive distributional equations" have been studied extensively in [3], [4], and a number of examples where symmetry breaking occurs for some values of a parameter corresponding to our λ were studied in [7], [15], [43].…”