Chromatic zeros on hierarchical lattices and equidistribution on parameter space

Abstract

Associated to any finite simple graph $\\Gamma$ is the chromatic polynomial $\\mathcal{P}\\Gamma(q)$ whose complex zeros are called the \_chromatic zeros of $\\Gamma$. A hierarchical lattice is a sequence of finite simple graphs ${\\Gamman}{n=0}^\\infty$ built recursively using a substitution rule expressed in terms of a generating graph. For each $n$, let $\\mun$ denote the probability measure that assigns a Dirac measure to each chromatic zero of $\\Gamma_n$. Under a mild hypothesis on the generating graph, we prove that the sequence $\\mu_n$ converges to some measure $\\mu$ as $n$ tends to infinity. We call $\\mu$ the \_limiting measure of chromatic zeros associated to ${\\Gamman}{n=0}^\\infty$. In the case of the diamond hierarchical lattice we prove that the support of $\\mu$ 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.