# 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 $Γ$ 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.

## 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