Associated to any finite simple graph Γ is the <i>chromatic polynomial </i>PΓ(q) whose complex zeros are called the <i>chromatic zeros </i>of Γ. A hierarchical lattice is a sequence of finite simple graphs {Γ<sub>n</sub>}∞<sub><i>n</i>-0</sub> built recursively using a substitution rule expressed in terms of a generating graph. For each <i>n</i>, let <i>μn</i> denote the probability measure that assigns a Dirac measure to each chromatic zero of Γ<sub><i>n</i></sub>. Under a mild hypothesis on the generating graph, we prove that the sequence <i>μn</i> converges to some measure <i>μ</i> as <i>n</i> tends to infinity. We call <i>μ</i> the limiting measure of <i>chromatic zeros</i> associated to {Γ<sub>n</sub>}∞<sub><i>n-</i>0</sub>. In the case of the Diamond Hierarchical Lattice we prove that the support of <i>μ</i> has Hausdorff dimension two.<div><br></div><div>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.<br></div>
| Identifer | oai:union.ndltd.org:purdue.edu/oai:figshare.com:article/12331136 |
| Date | 15 June 2020 |
| Creators | Ivan Chio (8422929) |
| Source Sets | Purdue University |
| Detected Language | English |
| Type | Text, Thesis |
| Rights | CC BY 4.0 |
| Relation | https://figshare.com/articles/Some_Connections_Between_Complex_Dynamics_and_Statistical_Mechanics/12331136 |
Page generated in 0.0948 seconds