Spatial mixing and approximate counting for Potts model on graphs with bounded average degree

We propose a notion of contraction function for a family of graphs and establish its connection to the strong spatial mixing for spin systems. More specifically, we show that for anti-ferromagnetic Potts model on families of graphs characterized by a specific contraction function, the model exhibits strong spatial mixing, and if further the graphs exhibit certain local sparsity which are very natural and easy to satisfy by typical sparse graphs, then we also have FPTAS for computing the partition function. This new characterization of strong spatial mixing of multi-spin system does not require maximum degree of the graphs to be bounded, but instead it relates the decay of correlation of the model to a notion of effective average degree measured by the contraction of a function on the family of graphs. It also generalizes other notion of effective average degree which may determine the strong spatial mixing, such as the connective constant, whose connection to strong spatial mixing is only known for very simple models and is not extendable to general spin systems. As direct consequences: (1) we obtain FPTAS for the partition function of $q$-state anti-ferromagnetic Potts model with activity $0\le\beta<1$ on graphs of maximum degree bounded by $d$ when $q> 3(1-\beta)d+1$, improving the previous best bound $\beta> 3(1-\beta)d$ and asymptotically approaching the inapproximability threshold $q=(1-\beta)d$, and (2) we obtain an efficient sampler (in the same sense of fully polynomial-time almost uniform sampler, FPAUS) for the Potts model on Erd\H{o}s-R\'enyi random graph $\mathcal{G}(n,d/n)$ with sufficiently large constant $d$, provided that $q> 3(1-\beta)d+4$. In particular when $\beta=0$, the sampler becomes an FPAUS for for proper $q$-coloring in $\mathcal{G}(n,d/n)$ with $q> 3d+4$, improving the current best bound $q> 5.5d$ for FPAUS for $q$-coloring in $\mathcal{G}(n,d/n)$.