Random Walks [Fall 2019]
Probability theory and either one of: Measure and Integration, Measure-theoretic probability.
Aim of the course
The course will treat random walks on the d-dimensional integer lattice. This topic is fundamental for researchers in probability theory and neighboring fields, both in itself and as a tool in an ample range of settings, such as interacting particle systems, random graphs and queuing theory. Special emphasis will be given to nearest-neighbor and finite range walks on Z^d, though more general increment distributions and walks on other graphs will also be touched upon. The course will begin with the classical theory, including gambler's ruin estimates for the one-dimensional walk, classification of random walks (periodicity, recurrence) and the local central limit theorem (using Fourier analysis and focusing on error estimates). Other topics to be included are: electrical networks, potential theory, Green function and heat kernel estimates, the construction of Brownian motion and relation to random walks. Time permitting, more advanced topics will be covered, such as: spanning trees, loop-erased walks, random walks in random environments, and random interlacements. This material will expose participants to a number of important tools of probability theory, such as couplings, martingales, stochastic domination and the second moment method.
Evgeny Verbitskiy, Daniel Valesin