For counting weighted independent sets weighted by a parameter $\lambda$ (known as the hard-core model) there is a beautiful connection between statistical physics phase transitions on infinite, regular trees and the computational complexity of approximate counting on graphs of maximum degree $D$. For $\lambda$ below the critical point there is an elegant algorithm devised by Weitz (2006). The drawback of his algorithm is that the running time is exponential in $\log D$. In this talk we’ll describe recent work which shows $O(n \log n)$ mixing time of the single-site Markov chain when the girth>6 and $D$ is at least a sufficiently large constant. Our proof utilizes BP (belief propagation) to design a distance function in our coupling analysis. This is joint work with C. Efthymiou, T. Hayes, D. Stefankovic, and Y. Yin, see: https://arxiv.org/abs/1604.01422