Some Connections Between Complex Dynamics and Statistical Mechanics

Associated to any finite simple graph $\Gamma$ is the {\em chromatic polynomial} \P_\Gamma(q) whose complex zeros are called the {\em 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 μ_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 μ_n converges to some measure μ as $n$ tends to infinity. We call μ the {\em 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 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.

Author dc:creator

• Chio, Ivan

Advisor dc:contributor.advisor

• Roeder, Roland K. W.

• Complex Dynamics
• Dynamical Systems
• Statistical Mechanics
• Hierarchical Lattices

Language dc:language.iso

en_US

Identifier URI

http://dx.doi.org/10.7912/C2/2412

OAI identifier oai:identifier

oai:scholarworks.indianapolis.iu.edu:1805/22848