Some Results On Two-Dimensional Lattice Random Field Models

Mateo Wirth, University of Pennsylvania


This dissertation deals with two different stochastic processes defined on the two-dimensional integer lattice. These are the random field Ising model with Gaussian disorder and the Gaussian free field. In each case the questions we study are different but throughout the goal is to understand the large-scale (macroscopic) behavior of the process. Likewise, the techniques used to analyze each model will be different, but throughout there will be an emphasis on geometric thinking and the properties of Gaussian processes. The results can be roughly summarized as follows\begin{itemize} \item The correlation length of the random field Ising model scales as $e^{\epsilon^{-4/3}}$ as the disorder strength $\epsilon$ goes to 0. \item For a metric Gaussian free field defined on a square of size $N$ and a macroscopic annulus inside that box, the length of the shortest crossing of the annulus by a path where the field is positive is at most $N (\log N)^{1/4}$. \item When defined on a rectangle, the metric graph and discrete Gaussian free fields exhibit different crossing probabilities. That is, the probability that there exists a path connecting the left and right sides of the box on which the field is positive differs between the two models. \end{itemize}