Difference: ProbabilityReadingGroup (13 vs. 14)

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META TOPICPARENT name="WebHome"

Probability Reading Group, Spring 2012

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  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...)
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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.
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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

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2/16 Dominos and the Gaussian Free Field  

Further topics TBA

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Tentative syllabus (as of 01/23/12)

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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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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 ( 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 learning about connections between critical spin models and discrete complex analysis (e.g. parafermions).
 

Starters

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

 
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