# Chromatic zeros on hierarchical lattices and equidistribution on parameter space

### Ivan Chio

University of Rochester, USA### Roland K.W. Roeder

Indiana University Purdue University Indianapolis, USA

## 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.

## Cite this article

Ivan Chio, Roland K.W. Roeder, Chromatic zeros on hierarchical lattices and equidistribution on parameter space. Ann. Inst. Henri Poincaré Comb. Phys. Interact. 8 (2021), no. 4, pp. 491–536

DOI 10.4171/AIHPD/109