Some Connections Between Complex Dynamics and Statistical Mechanics

Abstract

Associated to any finite simple graph Γ is the {\em chromatic polynomial} \PΓ(q) whose complex zeros are called the {\em chromatic zeros} of Γ. A hierarchical lattice is a sequence of finite simple graphs {Γn}n=0∞ built recursively using a substitution rule expressed in terms of a generating graph. For each n, let μn denote the probability measure that assigns a Dirac measure to each chromatic zero of Γn. Under a mild hypothesis on the generating graph, we prove that the sequence μn converges to some measure μ as n tends to infinity. We call μ the {\em limiting measure of chromatic zeros} associated to {Γn}n=0∞. In the case of the Diamond Hierarchical Lattice we prove that the support of μ has Hausdorff dimension two. The main techniques used come from holomorphic dynamics and more specifically the theories of activity/bifurcation currents and arithmetic dynamics. We prove a new equidistribution theorem that can be used to relate the chromatic zeros of a hierarchical lattice to the activity current of a particular marked point. We expect that this equidistribution theorem will have several other applications, and describe one such example in statistical mechanics about the Lee-Yang-Fisher zeros for the Cayley Tree.