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Probability Reading Group, Spring 2012

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

Schedule

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Date Topic/paper Presenter
2/3 Conformal invariance of domino tiling, Part I Joe Chen
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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
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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.
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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 function. We didn't actually prove convergence to GFF, but all the necessary technical lemmas have been covered, so please read on your own.
 
2/24 Discrete complex analysis Baris Ugurcan
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See Mercat; Kenyon; Chelkak-Smirnov.
3/1 (Th) Topics in potential theory Tianyi Zheng
3/2 (Fr) is prospie visit day, so the meeting is tentatively moved to Th, time TBA.
3/9 Hausdorff dimension of planar Brownian motion? Mark Cerenzia?
See Lawler-Schramm-Werner.
3/16 No meeting
Day before spring break
3/23 No meeting
Actual spring break
3/30 Meeting resumes?
Though Joe will be at the Statistical Mechanics & Conformal Invariance workshop at MSRI, he promises to bring home foods for thought.
 
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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).

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  June 4-29: PIMS Probability Summer School, University of British Columbia, Vancouver, BC. Application is closed.
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June 18-29: St. Petersburg Summer School in Probability & Statistical Physics, Chebyshev Laboratory, St. Petersburg, Russia. Application is closed.
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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.
 
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