Probability Reading Group, Spring 2012

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Topic: Statistical mechanics on discrete graphs and its scaling limits

Time & space coordinates: Fridays 2:30 - 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
Clarified the connection between the coupling function and the Green's function, and showed how $F_0, F_1$ enter into the moment formula of the height variation. We didn't have time to prove convergence to GFF, but all the necessary technical lemmas have been covered, so please read on your own.
2/24 Discrete complex analysis on isoradial graphs Baris Ugurcan
Covered notions of harmonicity and holomorphicity on isoradial graphs, and showed how a sequence of uniformly bounded discrete harmonic functions on finer isoradial graph approximations converge in subsequence to a harmonic function in the continuum.
3/2 No meeting
Prospie visit day
3/9 Topics in potential theory Tianyi Zheng
 
3/16 No meeting
Day before spring break
3/23 No meeting
Actual spring break
3/30 Hausdorff dimension of planar Brownian motion Mark Cerenzia
See Lawler-Schramm-Werner. Some stochastic calculus will be covered.

Tentative syllabus (which is soon becoming outdated)

We will start with Kenyon's proof 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 pending interest. Then we move onto a general survey of abelian spin models, as a preparation for discussing connections between critical spin models, duality, and order-disorder variable pairing (e.g. parafermions, bosonization).

Relevant papers covered (so far & soon)

Conformal invariance of domino tilings and convergence of height variation to GFF

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

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

Discrete complex analysis

Christian Mercat, Discrete Riemann Surfaces and the Ising Model. Comm. Math. Phys. 218, 177-216 (2001). SpringerLink arXiv

Richard Kenyon, The Laplacian and Dirac operators on critical planar graphs. Inventiones Mathematicae. 150, 409-439 (2002). SpringerLink arXiv

Dmitry Chelkak and Stanislav Smirnov, Discrete complex analysis on isoradial graphs. arXiv

Hausdorff dimension of planar Brownian motion

Gregory F. Lawler, Oded Schramm, and Wendelin Werner, The dimension of the planar Brownian frontier is $4/3$. Math. Res. Lett. 8, 401-411 (2001). MathJournals

(see also references within)

Related surveys

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

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

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

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

Gabor Pete's course "Critical phenomena and conformal invariance in the plane" at Budapest University of Technology and Economics: Spring 2012.

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. Application is closed.

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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