Random Walks and Their Scaling Limits

Abstract

This dissertation deals with two-dimensional random walks and their conformally invariant scaling limits. More precisely, we study two kinds of random walks: sum of independently identically distributed random variables (with Brownian motion as the scaling limit) and self-interacting random walks (with Schramm-Loewner Evolution processes as their scaling limits). We organize our main results in three parts. In the first part, we study two types of probability measures on Brownian paths: fixed time ensemble and fixed endpoints ensemble. We prove a relationship between those two ensembles. The relationship is that if we take a curve from the fixed time ensemble, weight it by a suitable power of the distance of the two endpoints and then apply the conformal map that takes those two endpoints to the endpoints from the other ensemble, then the resulting curve is distributed as the other ensemble. The second part deals with exploration processes and their scaling limits. We define two radial exploration processes in a domain D, i.e., self-interacting random walks between a boundary point and an interior point. We prove those processes satisfy the reversibility property. The reversibility property enables us to prove the distribution of the last hitting point with the boundary of any radial SLE₆ is the harmonic measure. We also define an exploration process in the full-plane and prove its scaling limit is the full-plane SLE₆. A by-product of these result is that the time-reversal of a radial SLE₆ trace in D (aiming at 0, say) after the last visit to ∂D is the full-plane SLE₆ trace (starting at 0) up to the first visit of ∂D. The last part is devoted to Dirichlet problems in a domain D. Let f be the solution to a continuous Dirichlet problem. We approximate the continuous Laplacian by the generator for a continuous-state random walk with each step uniformly distributed in a disk of radius h. Let f_h be the corresponding solution. We find the limit of (f_h-f)/h as h approaches 0.