Date of Award

Spring 1-1-2025

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

First Advisor

Kenyon, Richard

Abstract

We study multinomial percolation, which is bond percolation on a family of graphs ("blow-up graphs") defined in terms of a fixed graph G. We compute connection thresholds in terms of distances on G. Moreover, if the percolation probability is inversely proportional to the blow-up multiplicity, there is a phase transition above which a giant component emerges. We show that many properties of the giant component can be computed using an analytic function with variables indexed by the vertices of G. Moreover, we find that the giant component gives rise to a natural field on G. As the blow-up multiplicity grows, we show that this field converges to a Gaussian field, and its covariance is given by the square of a massive Green's function on G.The main tool to prove the results above is the combinatorics of trees on blowup graphs ("multinomial trees"). To this end, the first half of this thesis is dedicated to the theory of multinomial trees. This includes two parts. Enumeration results show that the asymptotic growth of such trees are given by relative entropies of certain distributions on G. Analytic results show that the multivariate exponential generating function for rooted multinomial trees has analytic properties that can be formulated in terms of a massive Laplacian operator on G.

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