We explore connections between the phenomenon of correlation decay and the
location of Lee-Yang and Fisher zeros for various spin systems. In particular
we show that, in many instances, proofs showing that weak spatial mixing on the
Bethe lattice (infinite Δ-regular tree) implies strong spatial mixing on
all graphs of maximum degree Δ can be lifted to the complex plane,
establishing the absence of zeros of the associated partition function in a
complex neighborhood of the region in parameter space corresponding to strong
spatial mixing. This allows us to give unified proofs of several recent results
of this kind, including the resolution by Peters and Regts of the Sokal
conjecture for the partition function of the hard core lattice gas. It also
allows us to prove new results on the location of Lee-Yang zeros of the
anti-ferromagnetic Ising model.
We show further that our methods extend to the case when weak spatial mixing
on the Bethe lattice is not known to be equivalent to strong spatial mixing on
all graphs. In particular, we show that results on strong spatial mixing in the
anti-ferromagnetic Potts model can be lifted to the complex plane to give new
zero-freeness results for the associated partition function. This extension
allows us to give the first deterministic FPTAS for counting the number of
q-colorings of a graph of maximum degree Δ provided only that q≥2Δ. This matches the natural bound for randomized algorithms obtained by
a straightforward application of Markov chain Monte Carlo. We also give an
improved version of this result for triangle-free graphs