We study the hard-core (gas) model defined on independent sets of an input graph where the independent sets are weighted by a parameter (aka fugacity) λ > 0. For constant Δ, previous work of Weitz (2006) established an FPTAS for the partition function for graphs of maximum degree Δ when λλc(Δ). The threshold λc(Δ) is the critical point for the statistical physics phase transition for uniqueness/non-uniqueness on the infinite Δ-regular tree. The running time of Weitz's algorithm is exponential in logΔ. Here we present an FPRAS for the partition function whose running time is O*(n2). We analyze the simple single-site Markov chain known as the Glauber dynamics for sampling from the associated Gibbs distribution. We prove there exists a constant Δ0 such that for all graphs with maximum degree Δ ≥ Δ0 and girth ≥ 7 (i.e., no cycles of length ≤ 6), the mixing time of the Glauber dynamics is O(n log n) when λ
['Charilaos Efthymiou', 'Thomas P. Hayes', 'Daniel Štefankovič', 'Eric Vigoda', 'Yitong Yin']
Convergence of MCMC and Loopy BP in the Tree Uniqueness Region for the Hard-Core Model