Improved Mixing Condition on the Grid for Counting and Sampling Independent Sets

Abstract

The hard-core model has received much attention in the past couple of decades as a lattice gas model with hard constraints in statistical physics, a multicast model of calls in communication networks, and as a weighted independent set problem in combinatorics, probability and theoretical computer science.In this model, each independent set I in a graph G is weighted proportionally to λ |I| , for a positive real parameter λ. For large λ, computing the partition function (namely, the normalizing constant which makes the weighting a probability distribution on a finite graph) on graphs of maximum degree ∆ ≥ 3, is a well known computationally challenging problem. More concretely, let λ c (T ∆ ) denote the critical value for the so-called uniqueness threshold of the hard-core model on the infinite ∆-regular tree; recent breakthrough results of Dror Weitz (2006) and Allan Sly (2010) have identified λ c (T ∆ ) as a threshold where the hardness of estimating the above partition function undergoes a computational transition.We focus on the well-studied particular case of the square lattice Z 2 , and provide a new lower bound for the uniqueness threshold, in particular taking it well above λ c (T 4 ). Our technique refines and builds on the tree of self-avoiding walks approach of Weitz, resulting in a new technical sufficient criterion (of wider applicability) for establishing strong spatial mixing (and hence uniqueness) for the hard-core model. Our new criterion achieves better bounds on strong spatial mixing when the graph has extra structure, improving upon what can be achieved by just using the maximum degree. Applying our technique to Z 2 we prove that strong spatial mixing holds for all λ < 2.3882, improving upon the work of Weitz that held for λ < 27/16 = 1.6875. Our results imply a fully-polynomial deterministic approximation algorithm for estimating the partition function, as well as rapid mixing of the associated Glauber dynamics to sample from the hard-core distribution.

Improved Bounds on the Phase Transition for the Hard-Core Model in 2-Dimensions

“…Consequently, they establish that λ c (Z 2 ) > 2.388. In addition, a recent paper [15] presents a simpler condition for establishing SSM based on the connective constant of Z 2 , but the bounds obtained by that approach are currently weaker than [13]. Our first result establishes a limit to these approaches by showing that SSM does not hold on T saw (Z 2 ).…”

Counting hypergraph matchings up to uniqueness threshold

“…The D and K are transition matrices from vertex-types to hyperedge-types and vice versa in T k,d . The definition can be seen as a hypergraph generalization of the branching matrix for multi-type Galton-Watson tree [27]. Since types (orbits) are invariant under all automorphisms from G, it is clear that the above D and K are well-defined for every automorphism group G on T k,d with finitely many orbits.…”

Approximate capacities of two-dimensional codes by spatial mixing

“…Previously it is known that strong spatial mixing holds for the hard-square constraint [25] and there exists a polynomial-time approximation scheme (PTAS) for computing its capacity [17,14]. To analyze the spatial mixing of two-dimensional codes arising from the aforementioned constraints, we apply several techniques from the state of the art of counting algorithms and statistical physics, including: self-avoiding walk tree [25], sequential cavity method [6], branching matrix [18], the potential function proposed in [11], connective-constant-based strong spatial mixing [22], and the necessary condition for correlation decay in [23]. We make the following discoveries:…”

“…The self-avoiding walk tree was introduced in [25] to deal with Boolean-state pairwise constraints, and was generalized in [3] and [16] into its full-fledged power to deal with matching, and multi-state multi-wise constraints. These techniques were improved in a series of works [18,11,22,21,23].…”