The geometry of dimer models

David Cimasoni[1]

  • [1] Université de Genève, Section de mathématiques, 2-4 rue du Lièvre, 1211 Genève 4, Switzerland

Winter Braids Lecture Notes (2014)

  • Volume: 1, page 1-14
  • ISSN: ?? -

Abstract

top
This is an expanded version of a three-hour minicourse given at the winterschool Winterbraids IV held in Dijon in February 2014. The aim of these lectures was to present some aspects of the dimer model to a geometrically minded audience. We spoke neither of braids nor of knots, but tried to show how several geometric tools that we know and love (e.g. (co)homology, spin structures, real algebraic curves) can be applied to very natural problems in combinatorics and statistical physics. These lecture notes do not contain any new results, but give a (relatively original) account of the works of Kasteleyn [14], Cimasoni-Reshetikhin [4] and Kenyon-Okounkov-Sheffield [16].

How to cite

top

Cimasoni, David. "The geometry of dimer models." Winter Braids Lecture Notes 1 (2014): 1-14. <http://eudml.org/doc/275710>.

@article{Cimasoni2014,
abstract = {This is an expanded version of a three-hour minicourse given at the winterschool Winterbraids IV held in Dijon in February 2014. The aim of these lectures was to present some aspects of the dimer model to a geometrically minded audience. We spoke neither of braids nor of knots, but tried to show how several geometric tools that we know and love (e.g. (co)homology, spin structures, real algebraic curves) can be applied to very natural problems in combinatorics and statistical physics. These lecture notes do not contain any new results, but give a (relatively original) account of the works of Kasteleyn [14], Cimasoni-Reshetikhin [4] and Kenyon-Okounkov-Sheffield [16].},
affiliation = {Université de Genève, Section de mathématiques, 2-4 rue du Lièvre, 1211 Genève 4, Switzerland},
author = {Cimasoni, David},
journal = {Winter Braids Lecture Notes},
language = {eng},
pages = {1-14},
publisher = {Winter Braids School},
title = {The geometry of dimer models},
url = {http://eudml.org/doc/275710},
volume = {1},
year = {2014},
}

TY - JOUR
AU - Cimasoni, David
TI - The geometry of dimer models
JO - Winter Braids Lecture Notes
PY - 2014
PB - Winter Braids School
VL - 1
SP - 1
EP - 14
AB - This is an expanded version of a three-hour minicourse given at the winterschool Winterbraids IV held in Dijon in February 2014. The aim of these lectures was to present some aspects of the dimer model to a geometrically minded audience. We spoke neither of braids nor of knots, but tried to show how several geometric tools that we know and love (e.g. (co)homology, spin structures, real algebraic curves) can be applied to very natural problems in combinatorics and statistical physics. These lecture notes do not contain any new results, but give a (relatively original) account of the works of Kasteleyn [14], Cimasoni-Reshetikhin [4] and Kenyon-Okounkov-Sheffield [16].
LA - eng
UR - http://eudml.org/doc/275710
ER -

References

top
  1. Cahit Arf. Untersuchungen über quadratische Formen in Körpern der Charakteristik 2. I. J. Reine Angew. Math., 183:148–167, 1941. Zbl0025.01403MR8069
  2. Michael F. Atiyah. Riemann surfaces and spin structures. Ann. Sci. École Norm. Sup. (4), 4:47–62, 1971. Zbl0212.56402MR286136
  3. David Cimasoni. Dimers on graphs in non-orientable surfaces. Lett. Math. Phys., 87(1-2):149–179, 2009. Zbl1167.82006MR2480651
  4. David Cimasoni and Nicolai Reshetikhin. Dimers on surface graphs and spin structures. I. Comm. Math. Phys., 275(1):187–208, 2007. Zbl1135.82006MR2335773
  5. David Cimasoni and Nicolai Reshetikhin. Dimers on surface graphs and spin structures. II. Comm. Math. Phys., 281(2):445–468, 2008. Zbl1168.82012MR2410902
  6. N. P. Dolbilin, Yu. M. Zinov’ev, A. S. Mishchenko, M. A. Shtan’ko, and M. I. Shtogrin. Homological properties of two-dimensional coverings of lattices on surfaces. Funktsional. Anal. i Prilozhen., 30(3):19–33, 95, 1996. Zbl0879.60115MR1435135
  7. Jack Edmonds. Optimum branchings. J. Res. Nat. Bur. Standards Sect. B, 71B:233–240, 1967. Zbl0155.51204MR227047
  8. Michael E. Fisher. Statistical mechanics of dimers on a plane lattice. Phys. Rev., 124(6):1664–1672, Dec 1961. Zbl0105.22403MR136399
  9. Anna Galluccio and Martin Loebl. On the theory of Pfaffian orientations. I. Perfect matchings and permanents. Electron. J. Combin., 6:Research Paper 6, 18 pp. (electronic), 1999. Zbl0909.05005MR1670286
  10. Axel Harnack. Ueber die Vieltheiligkeit der ebenen algebraischen Curven. Math. Ann., 10(2):189–198, 1876. MR1509883
  11. Dennis Johnson. Spin structures and quadratic forms on surfaces. J. London Math. Soc. (2), 22(2):365–373, 1980. Zbl0454.57011MR588283
  12. P. W. Kasteleyn. The statistics of dimers on a lattice. Physica, 27:1209–1225, 1961. Zbl1244.82014
  13. P. W. Kasteleyn. Dimer statistics and phase transitions. J. Mathematical Phys., 4:287–293, 1963. MR153427
  14. P. W. Kasteleyn. Graph theory and crystal physics. In Graph Theory and Theoretical Physics, pages 43–110. Academic Press, London, 1967. Zbl0205.28402MR253689
  15. Richard Kenyon and Andrei Okounkov. Planar dimers and Harnack curves. Duke Math. J., 131(3):499–524, 2006. Zbl1100.14047MR2219249
  16. Richard Kenyon, Andrei Okounkov, and Scott Sheffield. Dimers and amoebae. Ann. of Math. (2), 163(3):1019–1056, 2006. Zbl1154.82007MR2215138
  17. L. Lovász and M. D. Plummer. Matching theory, volume 121 of North-Holland Mathematics Studies. North-Holland Publishing Co., Amsterdam, 1986. Annals of Discrete Mathematics, 29. Zbl0618.05001MR859549
  18. G. Mikhalkin. Real algebraic curves, the moment map and amoebas. Ann. of Math. (2), 151(1):309–326, 2000. Zbl1073.14555MR1745011
  19. Grigory Mikhalkin and Hans Rullgård. Amoebas of maximal area. Internat. Math. Res. Notices, (9):441–451, 2001. Zbl0994.14032MR1829380
  20. H. N. V. Temperley and Michael E. Fisher. Dimer problem in statistical mechanics—an exact result. Philos. Mag. (8), 6:1061–1063, 1961. Zbl0126.25102MR136398
  21. Glenn Tesler. Matchings in graphs on non-orientable surfaces. J. Combin. Theory Ser. B, 78(2):198–231, 2000. Zbl1025.05052MR1750896
  22. L. G. Valiant. The complexity of computing the permanent. Theoret. Comput. Sci., 8(2):189–201, 1979. Zbl0415.68008MR526203

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.