Chromatic zeros on hierarchical lattices and equidistribution on parameter space

Chio, Ivan; Roeder, Roland K.W.

Annales de l’Institut Henri Poincaré D, Tome 8 (2021) no. 4, pp. 491-536.

Associated to any finite simple graph Γ is the chromatic polynomial 𝒫 Γ (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 μ n 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 μ n 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 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.

Classification : 82-XX, 05-XX, 37-XX
Mots-clés : Chromatic zeros, holomorphic dynamics, activity current, equidistribution problems, Potts model