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Some Connections Between Complex Dynamics and Statistical Mechanics

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posted on 15.06.2020 by Ivan Chio
Associated to any finite simple graph Γ is the chromatic polynomial PΓ(q) whose complex zeros are called the chromatic zeros of Γ. A hierarchical lattice is a sequence of finite simple graphs {Γn}∞n-0 built recursively using a substitution rule expressed in terms of a generating graph. For each n, let μn denote the probability measure that assigns a Dirac measure to each chromatic zero of Γn. Under a mild hypothesis on the generating graph, we prove that the sequence μn converges to some measure μ as n tends to infinity. We call μ the limiting measure of chromatic zeros associated to {Γn}∞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 anew equidistribution theorem that can be used to relate the chromatic zeros of ahierarchical lattice to the activity current of a particular marked point. We expect that this equidistribution theorem will have several other applications, and describe one such example in statistical mechanics about the Lee-Yang-Fisher zeros for the Cayley Tree.

History

Degree Type

Doctor of Philosophy

Department

Mathematics

Campus location

Indianapolis

Advisor/Supervisor/Committee Chair

ROLAND K. ROEDER

Additional Committee Member 2

MICHAL MISIUREWICZ

Additional Committee Member 3

RODRIGO A. PEREZ

Additional Committee Member 4

MAXIM L. YATTSELEV

Licence

Exports