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 $\{{\Gamma }_n{\}}_{n=0}^{\infty }$ built recursively using a substitution rule expressed in terms of a generating graph. For each $n$, let ${\mu }_n$ 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 $\{{\Gamma }_n{\}}_{n=0}^{\infty }$. 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.