Convergence of MCMC and Loopy BP in the Tree Uniqueness Region for the Hard-Core Model

We study the hard-core model defined on independent sets of an input graph where the independent sets are weighted by a parameter $\lambda >0$. For constant $\mathrm{\Delta }$, previous work of Weitz (2006) established an FPTAS for the partition function for graphs of maximum degree $\mathrm{\Delta }$ when $\lambda< \lambda_c(\Delta)$\mathrm{.}Thethreshold$\lambda_c(\Delta)$ is the critical point for the phase transition for uniqueness/non-uniqueness on the infinite $\mathrm{\Delta }$-regular trees. Sly (2010) showed that there is no FPRAS, unless NP=RP, when $\lambda >{\lambda }_c(\mathrm{\Delta })$. The running time of Weitz’s algorithm is exponential in $\log (\mathrm{\Delta })$. Here we present an FPRAS for the partition function whose running time is $O^{\ast }(n^2)$. We analyze the simple single-site Glauber dynamics for sampling from the associated Gibbs distribution. We prove there exists a constant ${\mathrm{\Delta }}_0$ such that for all graphs with maximum degree $\mathrm{\Delta }\ge {\mathrm{\Delta }}_0$ and girth $\ge 7$, the mixing time of the Glauber dynamics is $O(n\log (n))$ when $\lambda <{\lambda }_c(\mathrm{\Delta })$. Our work complements that of Weitz which applies for constant $\mathrm{\Delta }$ whereas our work applies for all $\mathrm{\Delta }\ge {\mathrm{\Delta }}_0$. We utilize loopy BP (belief propagation), a widely-used inference algorithm. 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 converges, after a short burn-in period, 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 $\geq 6$and$\lambda<\lambda_c(\Delta)$.