Some Connections between Complex Dynamics and Statistical Mechanics

Ivan Chio, Purdue University

Abstract

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 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, and describe one such example in statistical mechanics about the Lee-Yang-Fisher zeros for the Cayley Tree.

Degree

Ph.D.

Advisors

Roeder, Purdue University.

Subject Area

Statistical physics|Mathematics|Physics

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