Chromatic zeros on hierarchical lattices and equidistribution on parameter space
| Autor: | Roland K. W. Roeder, Ivan Chio |
|---|---|
| Rok vydání: | 2021 |
| Předmět: |
Statistics and Probability
Algebra and Number Theory Dirac measure Zero (complex analysis) Statistical and Nonlinear Physics Equidistribution theorem Chromatic polynomial 16. Peace & justice Arithmetic dynamics Measure (mathematics) Combinatorics symbols.namesake Hausdorff dimension symbols Discrete Mathematics and Combinatorics Geometry and Topology Probability measure Mathematics |
| Zdroj: | Annales de l’Institut Henri Poincaré D. 8:491-536 |
| ISSN: | 2308-5827 |
| DOI: | 10.4171/aihpd/109 |
| Popis: | Associated to any finite simple graph $\Gamma$ is the chromatic polynomial $P_\Gamma(q)$ whose complex zeroes 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. |
| Databáze: | OpenAIRE |
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