Correlation Decay and Partition Function Zeros: Algorithms and Phase Transitions

We explore connections between the phenomenon of correlation decay (more precisely, strong spatial mixing) and location Lee--Yang Fisher zeros for various spin systems. In particular we show that, in many instances, proofs showing that weak mixing on Bethe lattice (infinite $\Delta$-regular tree) implies all graphs maximum degree $\Delta$ can be lifted to complex plane, establishing absence associated partition function a neighborhood region parameter space corresponding mixing. This allows us give unified several recent results this kind, including resolution by Peters Regts Sokal conjecture hard-core gas. It also prove new antiferromagnetic Ising model. further our methods extend case when is not known equivalent graphs. particular, Potts model plane zero-freeness function, significantly sharpening previous others. extension independent algorithmic interest: it first polynomial time deterministic approximation algorithm (a fully scheme (FPTAS)) counting number $q$-colorings graph provided only $q\ge 2\Delta$, question has been studied intensively. matches natural bound randomized algorithms obtained straightforward application Markov chain Monte Carlo. triangle-free, applies under weaker condition $q \geq \alpha\Delta + \beta$, where $\alpha \approx 1.764$ $\beta = \beta(\alpha)$ are absolute constants.