Some Connections Between Complex Dynamics and Statistical Mechanics

2020-06-15T14:33:42Z (GMT) by Ivan Chio
Associated to any finite simple graph Γ is the chromatic polynomial PΓ(q) whose complex zeros are called the 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 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 anew equidistribution theorem that can be used to relate the chromatic zeros of ahierarchical 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.