Probability Reading Group, Spring 2012

Topic: Statistical mechanics on discrete graphs and its scaling limits

Time & place: TBA. Please fill out this scheduling form to help us set a time.

Description (as of Dec 2011): This informal reading group will be in some sense a continuation and/or elaboration of the materials given in the 2011 Cornell Probability Summer School. The focus will be on statistical mechanics models on discrete lattices or isoradial graphs in two dimensions, which include percolation, Ising/Potts spin models, height function associated with domino tilings, and quantum gravity models. We wish to understand their behavior in the scaling limit, that is, when either the size of the graph tends to infinity, or the underlying mesh radius goes to zero. Remarkably, many of these models at criticality converge to conformally invariant objects such as Schramm-Loewner evolution (SLE) curves or Gaussian free fields.

The goals of this seminar are to understand these discrete models and their limiting objects, and along the way learn the techniques used to prove convergence to the scaling limit.

Tentatively we plan to meet once a week for 60-90 minutes. The exact scope of topics covered will be announced prior to the spring semester, and may evolve according to the interests of the participants.

Prerequisites: Not afraid of measure-theoretic probability theory (as covered in MATH 6710-6720) and complex analysis (MATH 6120). No prior experience with statistical mechanics is needed.

For more information and expression of interest please contact Joe P. Chen (joe.p.chen@cornell.edu).

Prime surveys

Hugo Duminil-Copin and Stanislav Smirnov, Conformal invariance of lattice models. Lecture notes for the 2010 Clay Mathematical Institute Summer School. http://arxiv.org/abs/1109.1549

Geoffrey Grimmett, Three theorems in discrete random geometry. To be published in Probability Surveys (2011+). http://arxiv.org/abs/1110.2395

Richard Kenyon, Lectures on dimers. In Statistical Mechanics, IAS/Park City Mathematical Series 2007, S. Sheffield & T. Spencer, eds. (2009). (pdf)

Scott Sheffield, Gaussian free fields for mathematicians. Probability Theory & Related Fields 139, 521-541 (2007). http://arxiv.org/abs/math/0312099

Stanislav Smirnov, Discrete Complex Analysis and Probability. Proceedings of the International Congress of Mathematicians (ICM), Hyderabad, India (2010). http://arxiv.org/abs/1009.6077

Nike Sun, Conformally invariant scaling limits in planar critical percolation. Probability Surveys 8, 155-209 (2011). (pdf download)

Relevant links

MIT probability group reading seminar on 2D statistical physics: Fall 2010.

MIT Math 18.177: Topics in Stochastic processes, taught by Scott Sheffield. Fall 2009, Fall 2011.

Informal seminar on SLE at UC Berkeley, year unknown.

A collection of literatures by Pierre Nolin is here.

Info on 2012 summer schools

June 4-29: PIMS Probability Summer School, University of British Columbia, Vancouver, BC. Deadline: Dec 30, 2011.

June 18-29: St. Petersburg Summer School in Probability & Statistical Physics, Chebyshev Laboratory, St. Petersburg, Russia. Deadline: Feb 12, 2012.

July 16-27: Cornell Probability Summer School.