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.

**Grading:**

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. http://front.math.ucdavis.edu/1109.1549

Christophe Garban and Jeff Steif: *Lectures on noise sensitivity and percolation*. Clay Institute Summer School in Buzios, 2010. http://front.math.ucdavis.edu/1102.5761

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

Geoffrey Grimmett. *Probability on graphs*. Cambridge University Press, 2010. http://www.statslab.cam.ac.uk/~grg/books/pgs.html.

Wouter Kager and Bernard Nienhuis. *A guide to Stochastic Loewner Evolution and its applications*. http://front.math.ucdavis.edu/0312.4756

Russ Lyons with Yuval Peres. *Probability on trees and networks*. Book in preparation, to appear at Cambridge University Press. http://mypage.iu.edu/%7Erdlyons/prbtree/prbtree.html

Peter Mörters and Yuval Peres. *Brownian motion*. Cambridge University Press, 2010. http://people.bath.ac.uk/maspm/book.pdf

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. http://front.math.ucdavis.edu/math.PR/0602151

Wendelin Werner. *Random planar curves and Schramm-Loewner evolutions.* Saint-Flour Summer School, 2002. http://front.math.ucdavis.edu/math.PR/0303354

Wendelin Werner. *Lectures on two-dimensional critical percolation*. IAS Park City Summer School, 2007. http://front.math.ucdavis.edu/0710.0856