"Dimer model"의 두 판 사이의 차이
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| 15번째 줄: | 15번째 줄: | ||
* Kasteleyn matrix | * Kasteleyn matrix | ||
* height function | * height function | ||
| − | * | + | * spectral curve |
| + | * surface tension | ||
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| 23번째 줄: | 26번째 줄: | ||
<h5>physics motivation</h5> | <h5>physics motivation</h5> | ||
| − | * | + | * dimer configuration can be considered as the covering of the graph by pairs of fermions connected by an edge |
| 48번째 줄: | 51번째 줄: | ||
<h5>energy and weight systems</h5> | <h5>energy and weight systems</h5> | ||
| − | * define a weight | + | * define a weight function on the edges of the graph \gamma<br><math>w:E(\Gamma)\to \mathbb{R}_{\geq 0}</math><br> |
* For a dimer configuration D,<br><math>w(D)=\prod_{e\in D} w(e)</math><br> | * For a dimer configuration D,<br><math>w(D)=\prod_{e\in D} w(e)</math><br> | ||
* energy function<br><math>\epsilon:E(\Gamma)\to \mathbb{R}</math><br> | * energy function<br><math>\epsilon:E(\Gamma)\to \mathbb{R}</math><br> | ||
| 58번째 줄: | 61번째 줄: | ||
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2011년 8월 8일 (월) 09:25 판
introduction
- relation to Bethe ansatz http://staff.science.uva.nl/~nienhuis/tiles.pdf
- domino tiling
basic notions
- dimer configurations
- set of dimer configurations
- partition function
- Kasteleyn matrix
- height function
- spectral curve
- surface tension
physics motivation
- dimer configuration can be considered as the covering of the graph by pairs of fermions connected by an edge
Termperley equivalence
- spanning trees on \gamma rooted at x
- Dimers on D(\gamma)
Domino tiling and height function
- bipartite graph
energy and weight systems
- define a weight function on the edges of the graph \gamma
\(w:E(\Gamma)\to \mathbb{R}_{\geq 0}\) - For a dimer configuration D,
\(w(D)=\prod_{e\in D} w(e)\) - energy function
\(\epsilon:E(\Gamma)\to \mathbb{R}\) - For a dimer configuration D,
\(\epsilon(D)=\sum_{e\in D} \epsilon(e)\) - energy and weight function
\(w(e)=\exp (-\frac{\epsilon(e)}{T})\) - partition function
\(Z_{\Gamma}=\sum_{D\subset \Gamma} \prod_{e\in D} w(e)\)
memo
- http://www.math.brown.edu/~rkenyon/talks/
- http://www.umich.edu/~mctp/SciPrgPgs/events/2006/2006glsc/talks/hanany.pdf
- http://www.lif.univ-mrs.fr/~fernique/info/slides_csr.pdf
history
encyclopedia
- http://en.wikipedia.org/wiki/Domino_tiling
- http://en.wikipedia.org/wiki/Lozenge
- http://en.wikipedia.org/wiki/Gaussian_free_field
- http://en.wikipedia.org/wiki/
- http://www.scholarpedia.org/
- http://www.proofwiki.org/wiki/
- Princeton companion to mathematics(Companion_to_Mathematics.pdf)
books
- Statistical mechanics
- 2010년 books and articles
- http://gigapedia.info/1/statistical+mechanics
- http://gigapedia.info/1/dimer
- http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=
links
expositions
- http://www.ams.org/bookstore?fn=20&arg1=genint&item=HAPPENING-7
- dimer models for mathematicians
- Dimers, Amoebae and Limit shapes
- Dimers, the complex burgers equation, and curves inscribed in polygonsl
- The dimer model Richard Kenyon,
- Dimer Problems Richard Kenyon, 2005
- Gaussian free fields for mathematiciansn Scott Sheffield, 2003
- An introduction to the dimer model Richard Kenyon, 2003
- The dimer model in Statistical mechanics
- Dimers and Dominos James Propp, 1992
- pictures
articles
- Cimasoni, David, 와/과Nicolai Reshetikhin. 2007. “Dimers on surface graphs and spin structures. II”. 0704.0273 (4월 2). doi:doi:10.1007/s00220-008-0488-3. http://arxiv.org/abs/0704.0273.
- Exact solution of close-packed dimers on the kagome lattice
- Fa Wang, F. Y. Wu, 2006
- [1]http://www.physics.neu.edu/faculty/wu%20files/pdf/Wu217_PRE74_020104%28R%29.pdf
- Limit shapes and the complex burgers equation
- Richard Kenyon, Andrei Okounkov, 2005-7
- Richard Kenyon, Andrei Okounkov, 2005-7
- Planar dimers and Harnack curves
- Richard Kenyon, Andrei Okounkov, 2003-11
- Dimers and Amoebae
- Richard Kenyon, Andrei Okounkov, Scott Sheffield, 2003-11
- Dimers, Tilings and Trees
- A variational principle for domino tilings
- Cohn H., Kenyon R., Propp J. (2001), J. Amer. Math.Soc., 14, no.2, 297-346
- Richard Kenyon, The Annals of Probability Vol. 28, No. 2 (Apr., 2000), pp. 759-795
- The asymptotic determinant of the discrete Laplacian
- Richard Kenyon, Acta Mathematica Volume 185, Number 2, 239-286, 2000
- W. P. Thurston, Conway’s tiling groups, Amer. Math. Monthly 97 (1990), 757–773.
- Kasteleyn, P. W. 1963. Dimer Statistics and Phase Transitions. Journal of Mathematical Physics 4, no. 2: 287. doi:10.1063/1.1703953.
- Statistical Mechanics of Dimers on a Plane Lattice
- Michael E. Fisher , Phys. Rev. 124, 1664–1672 (1961)
- The statistics of dimers on a lattice. I. The number of dimer arrangements on a quadratic lattice
- Kasteleyn, P. W. (1961), Physica 27 (12): 1209–1225
- http://www.ams.org/mathscinet
- http://www.zentralblatt-math.org/zmath/en/
- http://arxiv.org/
- http://www.pdf-search.org/
- http://pythagoras0.springnote.com/
- http://math.berkeley.edu/~reb/papers/index.html
- http://dx.doi.org/10.1007/978-0-8176-4842-8_20
question and answers(Math Overflow)
blogs
- 구글 블로그 검색
- http://ncatlab.org/nlab/show/HomePage