Critical phenomena and conformal invariance in the plane
BMETE95MM23, Spring 2012

Lecturer: Gábor Pete

Contact info: here

Poster: in Hungarian

Lectures: Thu 12:15-14:00 (changed!), R510. No classes Feb 14 - Mar 6. Extra classes to make up for the missed ones: Tue 12:15-14:00, R513.

Exercise solutions totaling 8 pts (each exercise is worth 2^{number of its stars}). Here is the current problem list.
Understanding and presenting a research paper, done in pairs. Here is the list of papers. The presentation times are set to May 31, June 7, June 14, Thu 2-4 pm, H61, but these can be changed if needed.

Topics might include (at least as papers suggested for presentation or for further reading):
Random Cluster models (percolation, Ising and Potts models, Uniform Spanning Tree), infinite volume limits with free and wired boundary conditions
FKG and other correlation inequalities
Influence of bits on Boolean functions, Russo's formula, sharp thresholds
Basics of critical planar percolation: Russo-Seymour-Welsh and Harris-Kesten theorems, continuity of the percolation probability at p_c
Conformal invariance of critical percolation: Cardy's formula, Smirnov's theorem
Universality for planar random walks, conformal invariance of Brownian motion, the Gaussian Free Field
Discrete complex analysis
Domino tiling height function, Kasteleyn-Temperley-Fisher-technology, stating Kenyon's theorem on convergence to GFF
Smirnov's theorem on the conformal invariance of Ising-spin and Ising-FK models. RSW-type results
Self-avoiding walks; their number determined by Duminil-Copin and Smirnov
Wilson's algorithm to generate a Uniform Spanning Tree with loop-erased random walks
Definition of the Schramm-Loewner Evolution. Some basic properties using Bessel processes
Computing percolation and Ising critical exponents using SLE
Near-critical models: Kesten's scaling relation for percolation, Onsager versus pivotal points in the Ising model, Minimal Spanning Tree
Dynamical percolation: noise sensitivity, Fourier spectrum, exceptional times, Incipient Infinite Cluster
The mixing time of the Glauber dynamics for the Ising model
SLEs as level curves in GFF. Random planar maps, Liouville quantum gravity, KPZ relation

Prerequisites: Basics of probability theory and stochastic processes, e.g., martingales, Brownian motion. Basics of complex analysis, e.g., Schwarz' lemma and Riemann mapping theorem. Basics of PDE, e.g., Dirichlet problem for the Laplace equation. Background in Stochastic Calculus is helpful, but it should be possible to pick it up on the way. If you are missing some of the background listed, talk to me about how serious your gaps are.

The main lecture notes we will use or could be useful for background:

Hugo Duminil-Copin and Stas Smirnov. Conformal invariance of lattice models. Clay Institute Summer School in Buzios, 2010.

Christophe Garban and Jeff Steif: Lectures on noise sensitivity and percolation. Clay Institute Summer School in Buzios, 2010.

Geoffrey Grimmett. The random-cluster model. Springer, Berlin, 2006.

Geoffrey Grimmett. Probability on graphs. Cambridge University Press, 2010.

Wouter Kager and Bernard Nienhuis. A guide to Stochastic Loewner Evolution and its applications.

Russ Lyons with Yuval Peres. Probability on trees and networks. Book in preparation, to appear at Cambridge University Press.

Peter Mörters and Yuval Peres. Brownian motion. Cambridge University Press, 2010.

Gábor Pete. Probability and geometry on groups. Book in preparation. PGG.pdf

Oded Schramm. Conformally invariant scaling limits (an overview and a collection of problems). Talk at the ICM 2006.

Wendelin Werner. Random planar curves and Schramm-Loewner evolutions. Saint-Flour Summer School, 2002.

Wendelin Werner. Lectures on two-dimensional critical percolation. IAS Park City Summer School, 2007.