Chromatic zeros on hierarchical lattices and equidistribution on parameter space

Abstract

Associated to any finite simple graph Gamma is the chromatic polynomial mathcalPGamma(q) whose complex zeros are called the _chromatic zeros of Gamma. A hierarchical lattice is a sequence of finite simple graphs Missing open brace for superscript{\\Gamman}{n=0}^\\infty{\\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 Gamman. Under a mild hypothesis on the generating graph, we prove that the sequence mun converges to some measure mu as n tends to infinity. We call mu the _limiting measure of chromatic zeros associated to Missing open brace for superscript{\\Gamman}{n=0}^\\infty{\\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.