Linji Yang

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

For the hard-core lattice gas model defined on independent sets weighted by an activity λ, we study the critical activity λ c (ℤ2) for the uniqueness threshold on the 2-dimensional integer lattice ℤ2. The conjectured value of the critical activity is approximately 3.796. Until recently, the best lower bound followed from algorithmic results of Weitz (2006). Weitz presented an FPTAS for approximating the partition function for graphs of constant maximum degree Δ when \(\lambda 2.388. In this paper, we establish an upper bound for this approach, by showing that SSM does not hold on T saw(ℤ2) when λ > 3.4. We also present a refinement of the approach of Restrepo et al. which improves the lower bound to λ c (ℤ2) > 2.48.

Improved inapproximability results for counting independent sets in the hard-core model

We study the computational complexity of approximately counting the number of independent sets of a graph with maximum degree Δ. More generally, for an input graph G=V,E and an activity λ>0, we are interested in the quantity ZGλ defined as the sum over independent sets I weighted as wI=λ|I|. In statistical physics, ZGλ is the partition function for the hard-core model, which is an idealized model of a gas where the particles have non-negligible size. Recently, an interesting phase transition was shown to occur for the complexity of approximating the partition function. Weitz showed an FPAS for the partition function for any graph of maximum degree Δ when Δ is constant and λ 0. We remove the upper bound in the assumptions of Slys result for Δi¾?4,5, that is, we show that there does not exist efficient randomized approximation algorithms for all λ>λcTΔ for Δ=3 and Δi¾?6. Slys inapproximability result uses a clever reduction, combined with a second-moment analysis of Mossel, Weitz and Wormald which prove torpid mixing of the Glauber dynamics for sampling from the associated Gibbs distribution on almost every regular graph of degree Δ for the same range of λ as in Slys result. We extend Slys result by improving upon the technical work of Mossel et al., via a more detailed analysis of independent sets in random regular graphs.

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

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

Ferromagnetic Potts Model: Refined #BIS-hardness and Related Results

I

Improved mixing condition on the grid for counting and sampling independent sets

in a graph

Phase Transition for Glauber Dynamics for Independent Sets on Regular Trees

G

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

is weighted proportionally to

Phase transition for Glauber dynamics for independent sets on regular trees

\lambda^{|I|}

Phase transition for the mixing time of the Glauber dynamics for coloring regular trees

, for a positive real parameter

Ferromagnetic Potts Model: Refined #BIS-hardness and Related Results.

\lambda