Indiana University-Purdue University Indianapolis (IUPUI) / 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
$\mu_n$ 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 $\mu_n$ converges to some measure $\mu$ as $n$ tends
to infinity. We call $\mu$ 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 $\mu$ 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, and describe one such example in statistical mechanics about the Lee-Yang-Fisher zeros for the Cayley Tree.
| Identifer | oai:union.ndltd.org:IUPUI/oai:scholarworks.iupui.edu:1805/22848 |
| Date | 05 1900 |
| Creators | Chio, Ivan |
| Contributors | Roeder, Roland K. W., Misiurewicz, Michal, Perez, Rodrigo A., Yattselev, Maxim L. |
| Source Sets | Indiana University-Purdue University Indianapolis |
| Language | en_US |
| Detected Language | English |
| Type | Thesis |
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