Dimer model

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

• 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)
• 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

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

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

articles

• Cimasoni, David, and Nicolai Reshetikhin. “Dimers on Surface Graphs and Spin Structures. II.” Communications in Mathematical Physics 281, no. 2 (July 2008): 445–68. doi:10.1007/s00220-008-0488-3. http://arxiv.org/abs/0704.0273.
• Wang, Fa, and F. Y. Wu. “Exact Solution of Close-Packed Dimers on the Kagomé Lattice.” Physical Review E 75, no. 4 (April 19, 2007): 040105. doi:10.1103/PhysRevE.75.040105.
• [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