Abstract-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) > 0. . 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 0 such that for all graphs with maximum degree ≥ 0 and girth ≥ 7 (i.e., no cycles of length ≤ 6), the mixing time of the Glauber dynamics is ( log ) when < ( ). Our work complements that of Weitz which applies for small constant whereas our work applies for all at least a sufficiently large constant 0 (this includes depending on = | |). Our proof utilizes loopy BP (belief propagation) which is a widely-used algorithm for inference in graphical models. A novel aspect of our work is using the principal eigenvector for the BP operator to design a distance function which contracts in expectation for pairs of states that behave like the BP fixed point. We also prove that the Glauber dynamics behaves locally like loopy BP. As a byproduct we obtain that the Glauber dynamics, after a short burn-in period, converges close to the BP fixed point, and this implies that the fixed point of loopy BP is a close approximation to the Gibbs distribution. Using these connections we establish that loopy BP quickly converges to the Gibbs distribution when the girth ≥ 6 and < ( ).
A more recent result of REF shows that for any 0 < δ < 1, there is a ∆ 0 (δ) such that for any ∆ ≥ ∆ 0 (δ) and λ = (1 − δ)λ c (∆), the Glauber dynamics mixes in O(n log n) steps for graphs with maximum degree ∆ and girth ≥ 7.
Convergence of MCMC and Loopy BP in the Tree Uniqueness Region for the Hard-Core Model
2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)
2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)
Computer Science Mathematics