MATH 777: Simple Random Walk (Fall 2005)

Instructor: Gregory Lawler

Meeting Time & Room

This will be an introduction to simple random walk on the integer lattice. I will focus on the main tools that are used to establish rigorous results: combinatorics; generating functions (incl. characteristic function), stopping times and strong Markov property, martingales, coupling of paths, and coupling with Brownian motion. I will restrict my consideration to simple nearest neighbor random walk (both discrete and continuous time).

At the end of the course I will discuss some processes that are derived from simple random walk: loop-erased random and uniform spanning trees, intersection properties of random walks, relation to random matrices.

I recently gave another course on random walk. Not only will this course not assume the previous course, the intent is for this course to go slower than the pace I set in that course. Much of the complication in that course was caused by considerations of more general walks than just simple walks. I will only work with simple random walk in this course.

The prerequisite for the course is a standard graduate course in probability including discrete time martingales and an introduction to Brownian motion.

Last modified: March 30, 2005