Probability Reading Group, Spring 2012

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Topic: Statistical mechanics on discrete graphs and its scaling limits

Time & space coordinates: Fridays 3:00 - 4:30 (when BRB calls!), Malott 205.

Regular participants: Mark Cerenzia, Joe Chen, Baris Ugurcan, Tianyi Zheng

Description: 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 random quadrangulations of surfaces. 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, Gaussian free fields, or quantum gravity models.

The goals of this seminar are to understand these discrete models and their limiting objects, and equally important, to learn the techniques used to prove convergence to the scaling limit. As such the emphasis will be on reading the mathematical proofs, as opposed to learning about heuristics. (Though it is often possible to understand the heuristic idea behind a rigorous result...)

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. The evolving choice of topics will follow the philosophy (in the words of L. Gross): From each according to her/his taste. To each according to his/her interest.

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

Schedule

Date Topic/paper Presenter
2/3 Conformal invariance of domino tiling, Part I Joe Chen
Setting the scene: Introduction to domino tiling and the dimer model, height function, and Gaussian free fields. Stated (and clarified) Kenyon's theorems on conformal invariance of height variation, and the convergence of height variation to GFF.
2/10 Conformal invariance of domino tiling, Part II Joe Chen
Covered Kasteleyn matrix, coupling function (a Cauchy kernel), convergence of the coupling function to $F_0, F_1$, which transform analytically under conformal maps.
2/17 Dominos and the Gaussian Free Field Joe Chen
Will clarify the connection between the coupling function and the Green's function. The rest is to show how $F_0, F_1$ enter into the moment formula of the height function, from which convergence to GFF is proved.
2/24 Discrete complex analysis Baris Ugurcan

Further topics TBD: A session devoted to potential theory & harmonic measures may be in the offing.

Tentative syllabus

We will start with Kenyon's proof (Papers 1 & 2) that the zero-mean height function associated with domino tilings of planar domains converges to the corresponding Gaussian free field. This will serve as nice review of concepts such as the Temperleyan tilings and Kasteleyn matrices ( combinatorics), (discrete) complex analysis, and random fields ( probability). The latter topics can be reinforced through Papers 3 & 4, pending interest. Then we move onto a general survey of abelian spin models (Paper 5), as a preparation for discussing connections between critical spin models, duality, and order-disorder variable pairing (e.g. parafermions, bosonization).

Starters

1) Richard Kenyon, Conformal invariance of domino tiling. Ann. Probab. 28, 759-795 (2000). Euclid

2) Richard Kenyon, Dominos and the Gaussian Free Field. Ann. Probab. 29, 1128-1137 (2001). Euclid

3) Stanislav Smirnov, Discrete Complex Analysis and Probability. Proceedings of the International Congress of Mathematicians (ICM), Hyderabad, India (2010). ArXiv

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

5)# Julien Dubedat, Topics on abelian spin models and related problems. Probability Surveys 8, 374-402 (2011). Link

Related 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. Probability Surveys 8, 403-441 (2011). Link

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

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

[# indicate expanded lecture notes from the 2011 Cornell Probability Summer School.]

Templates for reading materials

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, both surveys and original papers, is here.

Info on 2012 summer schools

June 4-29: PIMS Probability Summer School, University of British Columbia, Vancouver, BC. Application is closed.

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

July 8-21: 42nd Probability Summer School in St. Flour, St. Flour, France. Registration opens in February.

July 16-27: Cornell Probability Summer School.

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