Chromatic zeros on hierarchical lattices and equidistribution on parameter space

Chio, Ivan; Roeder, Roland K.W.

doi:10.4171/aihpd/109

Geodesic

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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.

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Résumé

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 ${Γ_{n}}_{n = 0}^{\infty}$ 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 ${Γ_{n}}_{n = 0}^{\infty}$ . 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.

Accepté le : 2021-03-18
Publié le : 2021-09-24

MR Zbl

DOI : 10.4171/aihpd/109

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

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     title = {Chromatic zeros on hierarchical lattices and equidistribution on parameter space},
     journal = {Annales de l{\textquoteright}Institut Henri Poincar\'e D},
     pages = {491--536},
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     doi = {10.4171/aihpd/109},
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     zbl = {1479.82009},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/aihpd/109/}
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Chio, Ivan; Roeder, Roland K.W. Chromatic zeros on hierarchical lattices and equidistribution on parameter space. Annales de l’Institut Henri Poincaré D, Tome 8 (2021) no. 4, pp. 491-536. doi : 10.4171/aihpd/109. http://geodesic.mathdoc.fr/articles/10.4171/aihpd/109/

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