Difference: ProbabilityReadingGroup (24 vs. 25)

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Probability Reading Group, Spring 2012

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Tentative syllabus (which is soon becoming outdated)

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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 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 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

 
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Starters

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Discrete complex analysis
 
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1) Richard Kenyon, Conformal invariance of domino tiling. Ann. Probab. 28, 759-795 (2000). Euclid
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Christian Mercat, Discrete Riemann Surfaces and the Ising Model. Comm. Math. Phys. 218, 177-216 (2001). SpringerLink arXiv
 
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2) Richard Kenyon, Dominos and the Gaussian Free Field. Ann. Probab. 29, 1128-1137 (2001). Euclid
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Richard Kenyon, The Laplacian and Dirac operators on critical planar graphs. Inventiones Mathematicae. 150, 409-439 (2002). SpringerLink arXiv
 
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3) Stanislav Smirnov, Discrete Complex Analysis and Probability. Proceedings of the International Congress of Mathematicians (ICM), Hyderabad, India (2010). ArXiv
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Dmitry Chelkak and Stanislav Smirnov, Discrete complex analysis on isoradial graphs. arXiv
 
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4) Scott Sheffield, Gaussian free fields for mathematicians. Probability Theory & Related Fields 139, 521-541 (2007). http://arxiv.org/abs/math/0312099
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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)

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

Related surveys

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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

 
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