Dimer model

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

1. 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.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

• http://www.google.com/search?hl=en&tbs=tl:1&q=

 

 

related items[[Schramm–Loewner evolution (SLE)|]]

• basic thermodynamics & statistical mechanics
• Schramm–Loewner evolution (SLE)
• 픽의 정리(Pick's Theorem)

 

 

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

• http://ipht.cea.fr/statcomb2009/dimers/abstracts.html

 

 

expositions

• 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 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
  
  • http://research.microsoft.com/en-us/um/people/cohn/randomtilings.html

 

 

articles

• 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
• 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)

• http://mathoverflow.net/search?q=dimer
• http://mathoverflow.net/search?q=

 

 

blogs

• 구글 블로그 검색
  
  • http://blogsearch.google.com/blogsearch?q=dimer+model
• http://ncatlab.org/nlab/show/HomePage

 

 

 

• Detexify2 - LaTeX symbol classifier
• 수식표현 안내
• The On-Line Encyclopedia of Integer Sequences
• http://functions.wolfram.com/