CEU Electronic Theses and Dissertations, 2024
| Author | Hrušková, Aranka |
|---|---|
| Title | Topics in Graph Limits, Wasserstein Isometries, and Discrete Harmonic Functions |
| Summary | The thesis consists of three parts. The first, developing various topics in the theory of graph limits, forms the bulk of the work. It is based on four papers, of which two are published and two are about to be submitted. The two published papers prove that certain combinatorial constructions on infinite graphs with bounded maximum degree can be obtained by local algorithms, meaning in particular that their approximate versions can be built along any graph sequence converging to such an infinite graph. The subsequent two papers focus on graph convergence in the intermediate regime, that is in the case when the number of edges grows superlinearly but subquadratically in the number of vertices along a given graph sequence. The second part, based on a paper under review, addresses the question of existence of isometries of the set of probability measures on a metric space which do not arise from an isometry of the metric space itself. We focus on the particular case when the metric space is a sphere and show that in that scenario, there are no such exotic isometries. The third and newest part investigates phenomena related to random walks on countable discrete groups. We ask for which bounded functions on $\Gamma$ there is a probability measure on the group with respect to which the function is harmonic, conjecture that changing an originally $\mu$-harmonic function at exactly one point produces a non-harmonisable function, and prove the conjecture in a special case. After that, we also extend the classical theorem that the values of a harmonic function converge almost surely along the trajectory of a random walk to say that the random harmonic function obtained by shifting our coordinates by the trajectory of the random walk converges pointwise almost surely to a constant function. |
| Supervisor | Dávid Kunszenti-Kovács |
| Department | Mathematics PhD |
| Full text | https://www.etd.ceu.edu/2024/hruskova_aranka.pdf |
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