CEU Electronic Theses and Dissertations, 2020
| Author | Bencs, Ferenc |
|---|---|
| Title | Graphs, groups and measures |
| Summary | In this thesis, we will investigate some combinatorial and algebraic objects through polynomial or probability theory. In the first part, we will investigate a kind of generating function for certain combinatorial objects. In the second part, we will examine a probability generalization of normal subgroups (the invariant random subgroups) in some self-similar groups. In the first two chapters, we investigate the partition function of the hard-core model, the independence polynomial of a graph, and the partition function of the Potts model. Our main focus is to find regions on the complex plane, where for a certain graph family, the partition function in question doesn't contain any zeros. In the third chapter, we define the descent polynomial, and we prove some conjectures concerning its coefficient sequences. As a corollary, we describe some zero-free regions for the descent polynomial. In the last chapter, we work on invariant random subgroups in groups acting on rooted trees. Our main concern is Alt_f(T) being the group of finitary even automorphisms of the d-ary rooted tree T. We prove that a nontrivial ergodic IRS of Alt_f(T) that acts without fixed points on the boundary of T contains a level stabilizer, in particular, it is the random conjugate of a finite index subgroup, but there are continuum many distinct atomless ergodic IRSs. We apply our techniques to extend these results for certain branch groups. |
| Supervisor | Csikvári, Péter |
| Department | Mathematics PhD |
| Full text | https://www.etd.ceu.edu/2020/bencs_ferenc.pdf |
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