Random walks, networks and environment (STAT351, Spring 2014)

In this course we cover selected material about probability on trees and networks, random walk in random and non-random environments, percolation and related interacting particle systems.

The first theme shall be probability on trees, where we shall meet the Galton-Watson trees, electrical networks, random walks on graphs and tree-indexed processes. It shall serve us also for introducing the first and second moment methods, Nash-Williams inequalities, Green and Martin capacities as well as selected topics and tools from percolation theory, fractal geometry, concentration inequalities and large deviations.

For this theme we shall cover most of Sect. 1-10 from the 1997 St. Flour summer school lecture notes by Yuval Peres (or the expanded text Probability on Trees and Networks by Russ Lyons, with Yuval Peres).

Turning to deal with percolation and related interacting particle systems, the second theme consists of guest lectures by Prof. Sidoravicius on absence of infinite cluster for critical percolation on two-dimensional slabs (from recent preprint by Hugu Duminil-Copin, Vladas Sidoravicius and Vincent Tassion), and single guest lectures by Dr. Cabezas (May 6), and Dr. Ahlberg (May 13), expending on their probability seminars of May 5 and May 12, respectively, covering recent results about activated random walks (see preprint), and first passage percolation (see preprint), respectively.

Our last theme is a short introduction to the study of random walk in random environment, covering selected material (mostly Sec. 2.1) of Ofer Zeitouni's 2001 St. Flour summer school lecture notes.

Supplements if needed shall be taken from current literature or from texts on Percolation (Grimmett) and Fractal geometry (Falconer).

Prerequisites: Familiarity with probability theory and stochastic processes at the level of completing Stat310A/B or with measure theory at the level of Math205A.

Requirements: Registered participants shall give short presentations related to the course material. These will take place in our last three meetings (May 29, June 3 and June 5), the attendence of which is mandatory .

Meeting: Sequoia 200, TTh 11:00-12:15. First meeting, April 1st.

Instructor: Amir Dembo (to set a meeting, please e-mail amir@math.stanford.edu).

See also seminar for current activity in related areas.