UCLA

This dissertation studies the asymptotic behavior of two probabilistic models.It consists of two parts, one from the specific area of random surface models within the wider branch of statistical physics, and the other from the area of random graphs, and more specifically on random graphs built upon a (non-trivial) underlying geometry. In both cases, the random structure homogenizes as the system size tends to infinity.

The first model under study is $\Z$-valued graph homomorphismsfrom the lattice $\Z^d$. It is known that this model exhibits limit shapes: a graph homomorphism chosen uniformly at random subject to fixed boundary values will, with high probability, lie uniformly close to a certain limiting profile over the bulk of the lattice subset. We extend the limit shape result, as quantified via a variational principle, large deviations principle, and concentration inequality, to a new version of the model. In the new version, the uniform distribution over graph homomorphism is perturbed by a random potential. This illustrates the robustness of the results and the methods used to prove them.

The second model is long-range percolation on the lattice graph~$\Z^d$.This is a random graph that includes all nearest-neighbor edges in~$\Z^d$ plus a random selection of longer edges. Longer edges are included or excluded at random and independently, where the probability that the edge with endpoints~$x$ and~$y$ is included is asymptotic to~$\beta |x-y|^{-s}$ for some~$s\in(d,2d)$ and some~$\beta > 0$. We sharpen the best known asymptotics for the graph distance under this choice of edge inclusion probability. The proof is inspired by the recent work~\cite{BL19}, which introduced and studied a continuum analogue of the model, and the conclusion is similar to the corresponding result contained therein.