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

 

 

weight systems

define a weight function on the edges of the graph \gamma

 

 

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.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/
• http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=

 

 

expositions

• Gaussian free fields for mathematicians
  
  • Scott Sheffield, 2003
• An introduction to the dimer model
  
  • Richard Kenyon, 2003
• Lozenge Tiling
• pictures
  
  • http://research.microsoft.com/en-us/um/people/cohn/randomtilings.html
• http://members.unine.ch/beatrice.detiliere/Cours/Ecole_Doctorale.pdf
  

• Dimers and Dominos
  
  • James Propp, 1992

 

 

articles

• 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)

• 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

 

 

experts on the field

• http://arxiv.org/

 

 

links

• http://ipht.cea.fr/statcomb2009/dimers/abstracts.html
• Detexify2 - LaTeX symbol classifier
• 수식표현 안내
• The On-Line Encyclopedia of Integer Sequences
• http://functions.wolfram.com/