MAT 1128HS, WINTER/SPRING 2011
TOPICS IN PROBABILITY: PERCOLATION IN THE PLANE, Z^d, AND BEYOND

Email: gabor @ as you would guess from this homepage address.
Downtown office: ES 4146 (on the South wall of the Earth Sciences building, towards West)

Lectures: Tue 10-12 am (BA 4010), Thu 10-11 am (Room BA 6183). Since I will be away for about two weeks in March, there will be some make-up classes Thu 9-10, same location, starting on January 27.

Grading: Exercise solutions totaling 8 pts (each exercise is worth 2^{number of its stars}) are to be submitted. Here are the exercises, constantly updated (last update: April 18). Even if you don't need a grade, I recommend looking at the exercises, they contain valuable background information.

Summary: Percolation is the simplest model of statistical mechanics that exhibits phase transitions, and it has become central in modern probability theory. Presently, it comes in three, no..., four flavours, and the course will be an introduction to all of them:

1. Percolation on Z^d is a classical subject, see, e.g., the monograph by Grimmett (1999). For instance, renormalization and, for high d, the lace expansion to find critical exponents are two famous techniques. A recent example that we will discuss is the proof of the Alexander-Orbach conjecture for high d by Kozma and Nachmias: the spectral dimension of random walk inside the Incipient Infinite Cluster of critical percolation is 4/3.
2. In the plane, duality helps a lot, and quite magically, critical percolation turns out to have a conformally invariant scaling limit (proved partly by Smirnov, 2001). In fact, percolation was the main motivation for the introduction of the Schramm-Loewner Evolution (2000), one of the largest successes of probability theory since its existence: it proves predictions made by Conformal Field Theory, by translating questions on conformally invariant fractal curves in the plane to exercises in stochastic calculus. We will also discuss noise and dynamical sensitivity, for which critical percolation is a key example, using Fourier analysis on the hypercube.
3. Percolation on Cayley graphs of groups is the scene of a rich interplay between algebraic, geometric and probabilistic properties, such as amenability, unimodularity, finitely presentedness. This area was initiated by Benjamini and Schramm in 1996.
4. Percolation on finite graphs was started in 1960 with the Erdős-Rényi random graph model G(n,p). It is a central topic of probabilistic combinatorics, and is also important for the rest of percolation theory.

Prerequisite(s): Measure theory and Real Analysis is required, Complex Analysis should be taken at least simultaneously. From the Fall semester, Graduate Probability I is expected.  A background in Stochastic Calculus would be helpful.

A few online references to give some idea of the area:

Vincent Beffara and Vladas Sidoravicius (2006): Percolation theory (9 pages in the Encyclopedia of Mathematical Physics), http://front.math.ucdavis.edu/0507.5220

Itai Benjamini and Oded Schramm (1996): Percolation beyond Z^d, many questions and a few answers,
http://www.math.washington.edu/~ejpecp/ECP/viewarticle.php?id=1561&layout=abstract

Oded Schramm (2006): Conformally invariant scaling limits (an overview and a collection of problems), http://front.math.ucdavis.edu/math.PR/0602151

Gordon Slade (2002): Scaling limits and super-Brownian motion, http://www.ams.org/notices/200209/index.html

Wendelin Werner (2007): Lectures on two-dimensional critical percolation, http://front.math.ucdavis.edu/0710.0856

My notes on Probability and Geometry on Groups, and

Russ Lyons with Yuval Peres: Probability on trees and networks, book in preparation.

My PGG notes, the WW notes above, and

Christophe Garban and Jeff Steif: Lectures on noise sensitivity and percolation, lecture notes for the 2010 Clay Summer School in Buzios