Chromatic zeros on hierarchical lattices and equidistribution on parameter space

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.