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==expositions== | ==expositions== | ||
| − | + | * Cimasoni, David. “The Geometry of Dimer Models.” arXiv:1409.4631 [math-Ph], September 16, 2014. http://arxiv.org/abs/1409.4631. | |
* http://www.ams.org/bookstore?fn=20&arg1=genint&item=HAPPENING-7 | * http://www.ams.org/bookstore?fn=20&arg1=genint&item=HAPPENING-7 | ||
* dimer models for mathematicians | * dimer models for mathematicians | ||
| 127번째 줄: | 127번째 줄: | ||
* pictures | * pictures | ||
** http://research.microsoft.com/en-us/um/people/cohn/randomtilings.html | ** http://research.microsoft.com/en-us/um/people/cohn/randomtilings.html | ||
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==articles== | ==articles== | ||
2014년 9월 16일 (화) 19:29 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
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\(Z_{\Gamma}=\sum_{D\subset \Gamma} \prod_{e\in D} w(e)\)
fH
- P(z_1,z_2,w) if weights are positive real., then P=0 is a Harnack curve of genus
- g=|int(N)|
- P(z_0,z_2)=0 is harnack if the amoeba map is at most 2-to-1.
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
- Schramm–Loewner evolution (SLE)
- basic thermodynamics & statistical mechanics
- Schramm–Loewner evolution (SLE)
- Gaussian free field theory
encyclopedia
- http://en.wikipedia.org/wiki/Domino_tiling
- http://en.wikipedia.org/wiki/Lozenge
- http://en.wikipedia.org/wiki/Gaussian_free_field
links
expositions
- Cimasoni, David. “The Geometry of Dimer Models.” arXiv:1409.4631 [math-Ph], September 16, 2014. http://arxiv.org/abs/1409.4631.
- 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
- 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