# Some Connections Between Complex Dynamics and Statistical Mechanics

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 {Γ_{n}}∞_{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 Γ_{n}. 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 {Γ_{n}}∞_{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.