JournalsaihpdVol. 8, No. 4pp. 491–536

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
Chromatic zeros on hierarchical lattices and equidistribution on parameter space cover
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Abstract

Associated to any finite simple graph Γ\Gamma is the chromatic polynomial PΓ(q)\mathcal{P}_\Gamma(q) whose complex zeros are called the chromatic zeros of Γ\Gamma. A hierarchical lattice is a sequence of finite simple graphs {Γn}n=0\{\Gamma_n\}_{n=0}^\infty built recursively using a substitution rule expressed in terms of a generating graph. For each nn, let μn\mu_n denote the probability measure that assigns a Dirac measure to each chromatic zero of Γn\Gamma_n. Under a mild hypothesis on the generating graph, we prove that the sequence μn\mu_n converges to some measure μ\mu as nn tends to infinity. We call μ\mu the limiting measure of chromatic zeros associated to {Γn}n=0\{\Gamma_n\}_{n=0}^\infty. In the case of the diamond hierarchical lattice we prove that the support of μ\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