"Dimer model"의 두 판 사이의 차이
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| 44번째 줄: | 44번째 줄: | ||
| − | <h5>weight systems</h5> | + | <h5>energy and weight systems</h5> |
| − | * define a weight functionon the edges of the graph \gamma | + | * define a weight functionon 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 | + | * energy function<br><math>\epsilon:E(\Gamma)\to \mathbb{R}</math><br> |
| + | * For a dimer configuration D,<br><math>\epsilon(D)=\sum_{e\in D} \epsilon(e)</math><br> | ||
| + | * energy and weight function<br><math>w(e)=\exp (-\frac{\epsilon(e)}{T})</math><br> | ||
| + | * partition function<br><math>Z_{\Gamma}=\sum_{D\subset \Gamma} \prod_{e\in D} w(e)</math><br> | ||
| 125번째 줄: | 128번째 줄: | ||
<h5>expositions</h5> | <h5>expositions</h5> | ||
| + | * dimer models for mathematicians | ||
* [http://www.math.brown.edu/%7Erkenyon/papers/de2.pdf Dimer Problems]<br> | * [http://www.math.brown.edu/%7Erkenyon/papers/de2.pdf Dimer Problems]<br> | ||
** Richard Kenyon, 2005 | ** Richard Kenyon, 2005 | ||
2010년 10월 28일 (목) 14:26 판
introduction
- relation to Bethe ansatz http://staff.science.uva.nl/~nienhuis/tiles.pdf
- domino tilings http://www-math.mit.edu/~rstan/transparencies/tilings3.pdf
basic notions
- dimer configurations
- set of dimer configurations
- partition function
- Kasteleyn matrix
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 functionon 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
\(Z_{\Gamma}=\sum_{D\subset \Gamma} \prod_{e\in D} w(e)\)
mathematica code
- detk[m_, n_] :=
N[Product[
Product[2 Cos[(Pi*l)/(m + 1)] + 2 I*Cos[(Pi*k)/(n + 1)], {k, 1,
n}], {l, 1, m}], 10]
Z[m_, n_] := Round[Sqrt[Abs[detk[m, n]]]]
Z[8, 8]
memo
- 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/
- http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=
links
- http://ipht.cea.fr/statcomb2009/dimers/abstracts.html
- http://www.math.brown.edu/~rkenyon/papers/index.html
expositions
- dimer models for mathematicians
- Dimer Problems
- Richard Kenyon, 2005
- Gaussian free fields for mathematicians
- Scott Sheffield, 2003
- An introduction to the dimer model
- Richard Kenyon, 2003
- http://members.unine.ch/beatrice.detiliere/Cours/Ecole_Doctorale.pdf
- Dimers and Dominos
- James Propp, 1992
- pictures
articles
- http://arxiv.org/abs/cond-mat/0612573
- http://www.physics.neu.edu/faculty/wu%20files/pdf/Wu217_PRE74_020104%28R%29.pdf
- Dimers, Tilings and Trees
- 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
- Kasteleyn, P. W. (1961), "The statistics of dimers on a lattice. I. The number of dimer arrangements on a quadratic lattice", 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/BF02392811
question and answers(Math Overflow)
blogs
- 구글 블로그 검색
- http://ncatlab.org/nlab/show/HomePage