Difference: ProbabilityReadingGroup (1 vs. 35)

Revision 352012-04-12 - jpc64

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META TOPICPARENT name="WebHome"

Probability Reading Group, Spring 2012

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3/9 Time change, trace Dirichlet forms, and relation to elementary potential theory Tianyi Zheng
Defined time changed process induced by additive functionals and Revuz measures. Covered local times, inverse local times, classical 2nd Ray-Knight theorem, construction of Dirichlet form for time-changed process. Example: RBM on a simply connected planar domain as a time change of RBM on the unit disc under conformal transformation. Mostly drawn from Chapter 5 of Chen-Fukushima.
4/13 SLE & planar Brownian motion, Part I Mark Cerenzia
Changed:
<
<
Introducing Mandelbrot's hypothesis and setup.
>
>
Introducing Mandelbrot's conjecture that the Hausdorff dim of planar Brownian motion is 4/3, and setup.
 
4/20 Cover times, blanket times, and majorizing measures Tianyi Zheng
Based on Ding-Lee-Peres.
4/27 SLE & planar Brownian motion, Part II Mark Cerenzia
Changed:
<
<
Discusses the SLE machinery which enables the proof of Mandelbrot's hypothesis.
>
>
Discusses the SLE machinery which enables the proof of Mandelbrot's conjecture.
 

Relevant papers covered (so far & soon)

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Probability Reading Group, Spring 2012

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Covered notions of harmonicity and holomorphicity on isoradial graphs; discrete exponential function; and mode of convergence of discrete harmonic functions to a true harmonic function in the continuum.
3/9 Time change, trace Dirichlet forms, and relation to elementary potential theory Tianyi Zheng
Defined time changed process induced by additive functionals and Revuz measures. Covered local times, inverse local times, classical 2nd Ray-Knight theorem, construction of Dirichlet form for time-changed process. Example: RBM on a simply connected planar domain as a time change of RBM on the unit disc under conformal transformation. Mostly drawn from Chapter 5 of Chen-Fukushima.
Changed:
<
<
4/6 (Tentative) Dimer observables, parafermions, and Ising correlators Joe Chen
aka "What I learned at MSRI while on extended spring break."
4/13 SLE & planar Brownian motion Mark Cerenzia
See Lawler-Schramm-Werner. Some stochastic calculus will be covered.
TBA Cover times, blanket times, and majorizing measures Tianyi Zheng
>
>
4/13 SLE & planar Brownian motion, Part I Mark Cerenzia
Introducing Mandelbrot's hypothesis and setup.
4/20 Cover times, blanket times, and majorizing measures Tianyi Zheng
 
Based on Ding-Lee-Peres.
Added:
>
>
4/27 SLE & planar Brownian motion, Part II Mark Cerenzia
Discusses the SLE machinery which enables the proof of Mandelbrot's hypothesis.
 

Relevant papers covered (so far & soon)

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Probability Reading Group, Spring 2012

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aka "What I learned at MSRI while on extended spring break."
4/13 SLE & planar Brownian motion Mark Cerenzia
See Lawler-Schramm-Werner. Some stochastic calculus will be covered.
Added:
>
>
TBA Cover times, blanket times, and majorizing measures Tianyi Zheng
Based on Ding-Lee-Peres.
 

Relevant papers covered (so far & soon)

Revision 322012-03-30 - jpc64

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META TOPICPARENT name="WebHome"

Probability Reading Group, Spring 2012

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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 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; discrete exponential function; and mode of convergence of discrete harmonic functions to a true harmonic function in the continuum.
Deleted:
<
<
3/2 No meeting
Prospie visit day
 
3/9 Time change, trace Dirichlet forms, and relation to elementary potential theory Tianyi Zheng
Defined time changed process induced by additive functionals and Revuz measures. Covered local times, inverse local times, classical 2nd Ray-Knight theorem, construction of Dirichlet form for time-changed process. Example: RBM on a simply connected planar domain as a time change of RBM on the unit disc under conformal transformation. Mostly drawn from Chapter 5 of Chen-Fukushima.
Deleted:
<
<
3/16 No meeting
Day before spring break
3/23 No meeting
Actual spring break
3/30 SLE & planar Brownian motion, Part I Mark Cerenzia
See Lawler-Schramm-Werner. Some stochastic calculus will be covered.
 
4/6 (Tentative) Dimer observables, parafermions, and Ising correlators Joe Chen
aka "What I learned at MSRI while on extended spring break."
Changed:
<
<
4/13 SLE & planar Brownian motion, Part II Mark Cerenzia
>
>
4/13 SLE & planar Brownian motion Mark Cerenzia
See Lawler-Schramm-Werner. Some stochastic calculus will be covered.
 

Relevant papers covered (so far & soon)

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Probability Reading Group, Spring 2012

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Actual spring break
3/30 SLE & planar Brownian motion, Part I Mark Cerenzia
See Lawler-Schramm-Werner. Some stochastic calculus will be covered.
Changed:
<
<
Time TBA SLE & planar Brownian motion, Part II Mark Cerenzia
Time TBA (Tentative) Dimer observables, parafermions, and Ising correlators Joe Chen
>
>
4/6 (Tentative) Dimer observables, parafermions, and Ising correlators Joe Chen
aka "What I learned at MSRI while on extended spring break."
4/13 SLE & planar Brownian motion, Part II Mark Cerenzia
 

Relevant papers covered (so far & soon)

Line: 68 to 69
  Hausdorff dimension of planar Brownian motion
Changed:
<
<		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
>
>		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)
 
Changed:
<
<		(see also references within)
>
>		 Relation between dimer & Ising models

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

 

Related surveys

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

Deleted:
<
<		# 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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Probability Reading Group, Spring 2012

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Day before spring break
3/23 No meeting
Actual spring break
Changed:
<
<
3/30 Hausdorff dimension of planar Brownian motion Mark Cerenzia
>
>
3/30 SLE & planar Brownian motion, Part I Mark Cerenzia
 
See Lawler-Schramm-Werner. Some stochastic calculus will be covered.
Added:
>
>
Time TBA SLE & planar Brownian motion, Part II Mark Cerenzia
Time TBA (Tentative) Dimer observables, parafermions, and Ising correlators Joe Chen
 

Relevant papers covered (so far & soon)

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META TOPICPARENT name="WebHome"

Probability Reading Group, Spring 2012

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Covered notions of harmonicity and holomorphicity on isoradial graphs; discrete exponential function; and mode of convergence of discrete harmonic functions to a true harmonic function in the continuum.
3/2 No meeting
Prospie visit day
Changed:
<
<
3/9 Topics in potential theory Tianyi Zheng
 
>
>
3/9 Time change, trace Dirichlet forms, and relation to elementary potential theory Tianyi Zheng
Defined time changed process induced by additive functionals and Revuz measures. Covered local times, inverse local times, classical 2nd Ray-Knight theorem, construction of Dirichlet form for time-changed process. Example: RBM on a simply connected planar domain as a time change of RBM on the unit disc under conformal transformation. Mostly drawn from Chapter 5 of Chen-Fukushima.
 
3/16 No meeting
Day before spring break
3/23 No meeting
Line: 60 to 60
  Dmitry Chelkak and Stanislav Smirnov, Discrete complex analysis on isoradial graphs. arXiv
Added:
>
>		 Time change, Dirichlet forms, and more potential theory

Z.-Q. Chen and Masatoshi Fukushima, Symmetric Markov Processes, Time Changes, and Boundary Theory. Princeton University Press (2011).

  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

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Probability Reading Group, Spring 2012

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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
Changed:
<
<
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.
>
>
Covered notions of harmonicity and holomorphicity on isoradial graphs; discrete exponential function; and mode of convergence of discrete harmonic functions to a true harmonic function in the continuum.
 
3/2 No meeting
Prospie visit day
3/9 Topics in potential theory Tianyi Zheng
Line: 42 to 42
 
3/30 Hausdorff dimension of planar Brownian motion Mark Cerenzia
See Lawler-Schramm-Werner. Some stochastic calculus will be covered.
Deleted:
<
<

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

Line: 54 to 50
  Richard Kenyon, Dominos and the Gaussian Free Field. Ann. Probab. 29, 1128-1137 (2001). Euclid
Added:
>
>		Scott Sheffield, Gaussian free fields for mathematicians. Probability Theory & Related Fields 139, 521-541 (2007). arXiv
  Discrete complex analysis

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

Line: 72 to 70
  Stanislav Smirnov, Discrete Complex Analysis and Probability. Proceedings of the International Congress of Mathematicians (ICM), Hyderabad, India (2010). ArXiv
Deleted:
<
<		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

Revision 272012-02-24 - jpc64

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Probability Reading Group, Spring 2012

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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.
Changed:
<
<
2/24 Discrete complex analysis Baris Ugurcan
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.
>
>
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
Changed:
<
<
3/30 Meeting resumes?
Though Joe will be away at MSRI attending the Statistical Mechanics & Conformal Invariance workshop, he promises to bring home plenty of foods for thought.
>
>
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)

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Probability Reading Group, Spring 2012

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Topic: Statistical mechanics on discrete graphs and its scaling limits

Changed:
<
<		Time & space coordinates: Fridays 3:00 - 4:30 (when BRB calls!), Malott 205.
>
>		Time & space coordinates: Fridays 2:30 - 4:30 (when BRB calls!), Malott 205.
  Regular participants: Mark Cerenzia, Joe Chen, Baris Ugurcan, Tianyi Zheng

Revision 252012-02-18 - jpc64

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Probability Reading Group, Spring 2012

Line: 43 to 44
 

Tentative syllabus (which is soon becoming outdated)

Changed:
<
<		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).
>
>		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

 
Changed:
<
<

Starters

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

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

Related surveys

Added:
>
>		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

Revision 242012-02-18 - jpc64

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Probability Reading Group, Spring 2012

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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
Changed:
<
<
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.
>
>
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 Baris Ugurcan
See Mercat; Kenyon; Chelkak-Smirnov.
3/1 (Th) Topics in potential theory Tianyi Zheng
Line: 39 to 39
 
3/23 No meeting
Actual spring break
3/30 Meeting resumes?
Changed:
<
<
Though Joe will be at the Statistical Mechanics & Conformal Invariance workshop at MSRI, he promises to bring home foods for thought.
>
>
Though Joe will be away at MSRI attending the Statistical Mechanics & Conformal Invariance workshop, he promises to bring home plenty of foods for thought.
 
Changed:
<
<

Tentative syllabus

>
>

Tentative syllabus (which is soon becoming outdated)

  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).

Revision 232012-02-17 - jpc64

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Probability Reading Group, Spring 2012

Line: 19 to 19
 For more information or expression of interest please contact Joe Chen (joe.p.chen@cornell.edu).

Schedule

Changed:
<
<
>
>
 
Date Topic/paper Presenter
<-- -->
Sorted ascending
2/3 Conformal invariance of domino tiling, Part I Joe Chen
Line: 27 to 27
 
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
Changed:
<
<
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.
>
>
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
Added:
>
>
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.
 
Deleted:
<
<		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).

Line: 74 to 84
  June 4-29: PIMS Probability Summer School, University of British Columbia, Vancouver, BC. Application is closed.
Changed:
<
<		June 18-29: St. Petersburg Summer School in Probability & Statistical Physics, Chebyshev Laboratory, St. Petersburg, Russia. 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.

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Probability Reading Group, Spring 2012

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Templates for reading materials

Added:
>
>		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.

Line: 71 to 74
  June 4-29: PIMS Probability Summer School, University of British Columbia, Vancouver, BC. Application is closed.
Changed:
<
<		June 18-29: St. Petersburg Summer School in Probability & Statistical Physics, Chebyshev Laboratory, St. Petersburg, Russia. Deadline: Feb 12, 2012.
>
>		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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Probability Reading Group, Spring 2012

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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
Changed:
<
<
Covered Kasteleyn matrix, coupling function, convergence of the coupling function to $F_0, F_1$, which transform analytically under conformal maps.
>
>
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
Changed:
<
<
Will finish showing how $F_0, F_1$ enter into the moment formula of the height function. Then convergence to GFF
>
>
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.

Revision 202012-02-11 - jpc64

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Probability Reading Group, Spring 2012

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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
Changed:
<
<
2/17 Dominos and the Gaussian Free Field  
2/24 ? Discrete complex analysis Baris Ugurcan
>
>
Covered Kasteleyn matrix, coupling function, 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 finish showing how $F_0, F_1$ enter into the moment formula of the height function. Then convergence to GFF
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

Revision 192012-02-10 - jpc64

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Probability Reading Group, Spring 2012

Line: 6 to 6
 

Topic: Statistical mechanics on discrete graphs and its scaling limits

Changed:
<
<		Time & space coordinates: Fridays 2:30 - 4:30 (when BRB calls!), Malott 205.
>
>		Time & space coordinates: Fridays 3:00 - 4:30 (when BRB calls!), Malott 205.
  Regular participants: Mark Cerenzia, Joe Chen, Baris Ugurcan, Tianyi Zheng

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Probability Reading Group, Spring 2012

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2/17 Dominos and the Gaussian Free Field  
2/24 ? Discrete complex analysis Baris Ugurcan
Changed:
<
<		Further topics TBA
>
>		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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Date Topic/paper Presenter
2/3 Conformal invariance of domino tiling, Part I Joe Chen
Changed:
<
<
Setting the scene: Introduction to domino tiling and the dimer model, height function, Kasteleyn matrix, and discrete (or pre-)holomorphicity. The plan is to cover the first 4 sections and however much of Section 5 of Kenyon's paper.
2/10 Conformal invariance of domino tiling, Part II  
>
>
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
 
2/17 Dominos and the Gaussian Free Field  
Added:
>
>
2/24 ? Discrete complex analysis Baris Ugurcan
  Further topics TBA

Tentative syllabus

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Date Topic/paper Presenter
Changed:
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<
2/2 Conformal invariance of domino tiling, Part I Joe Chen
>
>
2/3 Conformal invariance of domino tiling, Part I Joe Chen
 
Setting the scene: Introduction to domino tiling and the dimer model, height function, Kasteleyn matrix, and discrete (or pre-)holomorphicity. The plan is to cover the first 4 sections and however much of Section 5 of Kenyon's paper.
Changed:
<
<
2/9 Conformal invariance of domino tiling, Part II  
2/16 Dominos and the Gaussian Free Field  
>
>
2/10 Conformal invariance of domino tiling, Part II  
2/17 Dominos and the Gaussian Free Field  
  Further topics TBA

Tentative syllabus

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

Schedule

Changed:
<
<
Date Topic/paper Discussion leader
2/2 Conformal invariance of domino tiling, Part I Joe Chen
2/9 Conformal invariance of domino tiling, Part II  
2/16 Dominos and the Gaussian Free Field  
>
>

Date Topic/paper Presenter
2/2 Conformal invariance of domino tiling, Part I Joe Chen
Setting the scene: Introduction to domino tiling and the dimer model, height function, Kasteleyn matrix, and discrete (or pre-)holomorphicity. The plan is to cover the first 4 sections and however much of Section 5 of Kenyon's paper.
2/9 Conformal invariance of domino tiling, Part II  
2/16 Dominos and the Gaussian Free Field  
  Further topics TBA

Tentative syllabus

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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...)
Changed:
<
<		 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.
>
>		 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

Line: 25 to 25
 
2/16 Dominos and the Gaussian Free Field  

Further topics TBA

Changed:
<
<

Tentative syllabus (as of 01/23/12)

>
>

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).

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

Schedule

Changed:
<
<		2/2 (Fri): Conformal invariance of domino tiling, Part I (Joe Chen)

2/9 (Fri): Conformal invariance of domino tiling, Part II

2/16 (Fri): Dominos and the Gaussian Free Field

>
>
Date Topic/paper Discussion leader
2/2 Conformal invariance of domino tiling, Part I Joe Chen
2/9 Conformal invariance of domino tiling, Part II  
2/16 Dominos and the Gaussian Free Field  
  Further topics TBA

Tentative syllabus (as of 01/23/12)

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Topic: Statistical mechanics on discrete graphs and its scaling limits

Changed:
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<		Tentative time & place: Fridays 2:30 - 4:30 (when BRB calls!), Malott 205.
>
>		Time & space coordinates: Fridays 2:30 - 4:30 (when BRB calls!), Malott 205.
 
Changed:
<
<		Regular participants: Mark Cerenzia, Joe Chen, Baris Ugurcan, Tianyi Zheng
>
>		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.

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  Tentative time & place: Fridays 2:30 - 4:30 (when BRB calls!), Malott 205.
Changed:
<
<		Regular participants: Joe Chen, Baris Ugurcan, Tianyi Zheng
>
>		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.

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Topic: Statistical mechanics on discrete graphs and its scaling limits

Changed:
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<		Tentative time & place: Fridays 2:30 - 4:30 (when BRB calls!), room to be confirmed.
>
>		Tentative time & place: Fridays 2:30 - 4:30 (when BRB calls!), Malott 205.

Regular participants: 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.
Line: 14 to 16
  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.
Changed:
<
<		For more information or expression of interest please contact Joe P. Chen (joe.p.chen@cornell.edu).
>
>		For more information or expression of interest please contact Joe Chen (joe.p.chen@cornell.edu).
 

Schedule

2/2 (Fri): Conformal invariance of domino tiling, Part I (Joe Chen)

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Topic: Statistical mechanics on discrete graphs and its scaling limits

Changed:
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<		Time & place: Fridays 2:30 - 4:30 (when BRB calls!), room to be confirmed.
>
>		Tentative time & place: Fridays 2:30 - 4:30 (when BRB calls!), room to be confirmed.
  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.

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Topic: Statistical mechanics on discrete graphs and its scaling limits

Changed:
<
<		Time & place: TBD after class schedules have settled down during the first week.
>
>		Time & place: Fridays 2:30 - 4:30 (when BRB calls!), room to be confirmed.
  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.
Line: 15 to 15
  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.

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

Added:
>
>

Schedule

2/2 (Fri): Conformal invariance of domino tiling, Part I (Joe Chen)

2/9 (Fri): Conformal invariance of domino tiling, Part II

2/16 (Fri): Dominos and the Gaussian Free Field

Further topics TBA

 

Tentative syllabus (as of 01/23/12)

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).

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Probability Reading Group, Spring 2012

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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
Changed:
<
<		5)* Julien Dubedat, Topics on abelian spin models and related problems. Probability Surveys 8, 374-402 (2011). Link
>
>		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

Changed:
<
<		*Geoffrey Grimmett, Three theorems in discrete random geometry. To be published in Probability Surveys (2011+). http://arxiv.org/abs/1110.2395
>
>		#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)

Changed:
<
<
[* indicate expanded lecture notes from the 2011 Cornell Probability Summer School.]
>
>
[# indicate expanded lecture notes from the 2011 Cornell Probability Summer School.]
 

Templates for reading materials

Line: 52 to 52
 

Info on 2012 summer schools

Changed:
<
<		June 4-29: PIMS Probability Summer School, University of British Columbia, Vancouver, BC. Deadline: Dec 30, 2011.
>
>		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.
Added:
>
>		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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Topic: Statistical mechanics on discrete graphs and its scaling limits

Changed:
<
<		Time & place: TBA. Please fill out this scheduling form to help us set a time.
>
>		Time & place: TBD after class schedules have settled down during the first week.
 
Changed:
<
<		Description (as of Dec 2011): 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 quantum gravity models. 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 or Gaussian free fields.
>
>		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...)
Deleted:
<
<		Tentatively we plan to meet once a week for 60-90 minutes. The exact scope of topics covered will be announced prior to the spring semester, and may evolve according to the interests of the participants.
  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.

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

Added:
>
>

Tentative syllabus (as of 01/23/12)

 
Changed:
<
<

Prime surveys

>
>		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

 
Changed:
<
<		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
>
>		1) Richard Kenyon, Conformal invariance of domino tiling. Ann. Probab. 28, 759-795 (2000). Euclid
 
Changed:
<
<		Geoffrey Grimmett, Three theorems in discrete random geometry. To be published in Probability Surveys (2011+). http://arxiv.org/abs/1110.2395
>
>		2) Richard Kenyon, Dominos and the Gaussian Free Field. Ann. Probab. 29, 1128-1137 (2001). Euclid
 
Changed:
<
<		Richard Kenyon, Lectures on dimers. In Statistical Mechanics, IAS/Park City Mathematical Series 2007, S. Sheffield & T. Spencer, eds. (2009). (pdf)
>
>		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

 
Changed:
<
<		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

 
Changed:
<
<		Stanislav Smirnov, Discrete Complex Analysis and Probability. Proceedings of the International Congress of Mathematicians (ICM), Hyderabad, India (2010). http://arxiv.org/abs/1009.6077
>
>		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. To be published in Probability Surveys (2011+). http://arxiv.org/abs/1110.2395

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)
Added:
>
>
[* indicate expanded lecture notes from the 2011 Cornell Probability Summer School.]
 

Templates for reading materials

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Probability Reading Group, Spring 2012

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  Description (as of Dec 2011): 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 quantum gravity models. 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 or Gaussian free fields.
Changed:
<
<		The goals of this seminar are to understand these discrete models and their limiting objects, and along the way learn the techniques used to prove convergence to the scaling limit.
>
>		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...)
  Tentatively we plan to meet once a week for 60-90 minutes. The exact scope of topics covered will be announced prior to the spring semester, and may evolve according to the interests of the participants.

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Probability Reading Group, Spring 2012

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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.
Changed:
<
<		For more information and expression of interest please contact Joe P. Chen (joe.p.chen@cornell.edu).
>
>		For more information or expression of interest please contact Joe P. Chen (joe.p.chen@cornell.edu).
 

Prime surveys

Line: 32 to 32
  Nike Sun, Conformally invariant scaling limits in planar critical percolation. Probability Surveys 8, 155-209 (2011). (pdf download)
Changed:
<
<

Relevant links

>
>

Templates for reading materials

  MIT probability group reading seminar on 2D statistical physics: Fall 2010.
Line: 40 to 40
  Informal seminar on SLE at UC Berkeley, year unknown.
Changed:
<
<		A collection of literatures by Pierre Nolin is here.
>
>		A collection of literatures by Pierre Nolin, both surveys and original papers, is here.
 

Info on 2012 summer schools

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  Time & place: TBA. Please fill out this scheduling form to help us set a time.
Changed:
<
<		Description (as of Dec 2011): 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. Geoff Grimmett's lecture notes would be a nice starting point, supplemented by the relevant literatures, such as his books on percolation & random cluster model, as well as recent works of Kenyon, Sheffield, Smirnov, etc.
>
>		Description (as of Dec 2011): 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 quantum gravity models. 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 or Gaussian free fields.
 
Changed:
<
<		We could also concentrate on models in two dimensions, where many discrete models at criticality are believed (by physicists) and now proved (by mathematicians) to enjoy conformal invariance. These include percolation, spin models, height function associated with domino tilings, and quantum gravity models. The limiting objects often turn out to be instances of Schramm-Loewner evolution (SLE) or Gaussian free field.
>
>		The goals of this seminar are to understand these discrete models and their limiting objects, and along the way learn the techniques used to prove convergence to the scaling limit.
 
Changed:
<
<		An important goal is to learn the techniques used to prove these hard and physically interesting results.

Tentatively we plan to meet once a week for 60-90 minutes. The exact scope of topics covered will be announced before the spring semester starts, and may evolve according to the interests of the participants.

>
>		Tentatively we plan to meet once a week for 60-90 minutes. The exact scope of topics covered will be announced prior to the spring semester, and may evolve according to the interests of the participants.
  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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Topic: Statistical mechanics on discrete graphs and its scaling limits

Changed:
<
<		Time & place: TBA
>
>		Time & place: TBA. Please fill out this scheduling form to help us set a time.
 
Changed:
<
<		Description (as of Dec 2011): 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. Geoff Grimmett's lecture notes (http://arxiv.org/abs/1110.2395) would be a nice starting point, supplemented by the relevant literatures, such as his books on percolation & random cluster model, as well as recent works of Kenyon, Sheffield, Smirnov, etc.
>
>		Description (as of Dec 2011): 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. Geoff Grimmett's lecture notes would be a nice starting point, supplemented by the relevant literatures, such as his books on percolation & random cluster model, as well as recent works of Kenyon, Sheffield, Smirnov, etc.
 
Changed:
<
<		We could also concentrate on models in two dimensions, where many discrete models at criticality are believed (by physicists) and now proved (by mathematicians) to enjoy conformal invariance. These include percolation, spin models, height function associated with domino tilings, and quantum gravity models. The limiting objects often turn out to be instances of SLE, Gaussian free field, etc.
>
>		We could also concentrate on models in two dimensions, where many discrete models at criticality are believed (by physicists) and now proved (by mathematicians) to enjoy conformal invariance. These include percolation, spin models, height function associated with domino tilings, and quantum gravity models. The limiting objects often turn out to be instances of Schramm-Loewner evolution (SLE) or Gaussian free field.
  An important goal is to learn the techniques used to prove these hard and physically interesting results.
Line: 26 to 26
  Geoffrey Grimmett, Three theorems in discrete random geometry. To be published in Probability Surveys (2011+). http://arxiv.org/abs/1110.2395
Changed:
<
<		Richard Kenyon, Lectures on dimers. In Statistical Mechanics, IAS/Park City Mathematical Series 2007, eds. S. Sheffield & T. Spencer (2009). http://www.math.brown.edu/~rkenyon/papers/dimerlecturenotes.pdf
>
>		Richard Kenyon, Lectures on dimers. In Statistical Mechanics, IAS/Park City Mathematical Series 2007, S. Sheffield & T. Spencer, eds. (2009). (pdf)
 
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<		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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>		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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<		Stanislav Smirnov. Discrete Complex Analysis and Probability. Proceedings of the International Congress of Mathematicians (ICM), Hyderabad, India (2010). http://arxiv.org/abs/1009.6077
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>		Stanislav Smirnov, Discrete Complex Analysis and Probability. Proceedings of the International Congress of Mathematicians (ICM), Hyderabad, India (2010). http://arxiv.org/abs/1009.6077
 
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<		Nike Sun, Conformally invariant scaling limits in planar critical percolation. Probability Surveys 8, 155-209 (2011). http://www.i-journals.org/ps/include/getdoc.php?id=726&article=180&mode=pdf
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>		Nike Sun, Conformally invariant scaling limits in planar critical percolation. Probability Surveys 8, 155-209 (2011). (pdf download)
 

Relevant links

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<		MIT probability group reading seminar on 2D statistical physics (Fall 2010): http://math.mit.edu/~asafnach/2dstatphysics.html.
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>		MIT probability group reading seminar on 2D statistical physics: Fall 2010.
 
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<		Informal seminar on SLE at UC Berkeley (year unknown): http://www.eve.ucdavis.edu/plralph/sle-seminar/.
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>		MIT Math 18.177: Topics in Stochastic processes, taught by Scott Sheffield. Fall 2009, Fall 2011.
 
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<		A collection of literatures by Pierre Nolin: http://cims.nyu.edu/~nolin/AdvancedTopics/References.pdf.
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>		Informal seminar on SLE at UC Berkeley, year unknown.

A collection of literatures by Pierre Nolin is here.

 

Info on 2012 summer schools

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<		June 4-29: PIMS Probability Summer School, University of British Columbia, Vancouver, BC. Deadline: Dec 30, 2011. http://www.math.ubc.ca/Links/ssprob12/
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>		June 4-29: PIMS Probability Summer School, University of British Columbia, Vancouver, BC. Deadline: Dec 30, 2011.
 
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<		June 18-29: St. Petersburg Summer School in Probability & Statistical Physics, Chebyshev Laboratory, St. Petersburg, Russia. Deadline: Feb 12, 2012. http://spspsp.chebyshev.spb.ru/
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>		June 18-29: St. Petersburg Summer School in Probability & Statistical Physics, Chebyshev Laboratory, St. Petersburg, Russia. Deadline: Feb 12, 2012.
 
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<		July 16-27: Cornell Probability Summer School. http://www.math.duke.edu/~rtd/CPSS2012/index.html
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>		July 16-27: Cornell Probability Summer School.

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META TOPICPARENT name="WebHome"

Probability Reading Group, Spring 2012

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Topic: Statistical mechanics on discrete graphs and its scaling limits

Time & place: TBA

Description (as of Dec 2011): 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. Geoff Grimmett's lecture notes (http://arxiv.org/abs/1110.2395) would be a nice starting point, supplemented by the relevant literatures, such as his books on percolation & random cluster model, as well as recent works of Kenyon, Sheffield, Smirnov, etc.

We could also concentrate on models in two dimensions, where many discrete models at criticality are believed (by physicists) and now proved (by mathematicians) to enjoy conformal invariance. These include percolation, spin models, height function associated with domino tilings, and quantum gravity models. The limiting objects often turn out to be instances of SLE, Gaussian free field, etc.

An important goal is to learn the techniques used to prove these hard and physically interesting results.

Tentatively we plan to meet once a week for 60-90 minutes. The exact scope of topics covered will be announced before the spring semester starts, and may evolve according to the interests of the participants.

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.

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

Prime 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. To be published in Probability Surveys (2011+). http://arxiv.org/abs/1110.2395

Richard Kenyon, Lectures on dimers. In Statistical Mechanics, IAS/Park City Mathematical Series 2007, eds. S. Sheffield & T. Spencer (2009). http://www.math.brown.edu/~rkenyon/papers/dimerlecturenotes.pdf

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

Stanislav Smirnov. Discrete Complex Analysis and Probability. Proceedings of the International Congress of Mathematicians (ICM), Hyderabad, India (2010). http://arxiv.org/abs/1009.6077

Nike Sun, Conformally invariant scaling limits in planar critical percolation. Probability Surveys 8, 155-209 (2011). http://www.i-journals.org/ps/include/getdoc.php?id=726&article=180&mode=pdf

Relevant links

MIT probability group reading seminar on 2D statistical physics (Fall 2010): http://math.mit.edu/~asafnach/2dstatphysics.html.

Informal seminar on SLE at UC Berkeley (year unknown): http://www.eve.ucdavis.edu/plralph/sle-seminar/.

A collection of literatures by Pierre Nolin: http://cims.nyu.edu/~nolin/AdvancedTopics/References.pdf.

Info on 2012 summer schools

June 4-29: PIMS Probability Summer School, University of British Columbia, Vancouver, BC. Deadline: Dec 30, 2011. http://www.math.ubc.ca/Links/ssprob12/

June 18-29: St. Petersburg Summer School in Probability & Statistical Physics, Chebyshev Laboratory, St. Petersburg, Russia. Deadline: Feb 12, 2012. http://spspsp.chebyshev.spb.ru/

July 16-27: Cornell Probability Summer School. http://www.math.duke.edu/~rtd/CPSS2012/index.html

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