Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
Quadri-tilings of the Plane
Download PDF
Download PDF
  • Published: 27 April 2006

Quadri-tilings of the Plane

  • Béatrice de Tilière1 

Probability Theory and Related Fields volume 137, pages 487–518 (2007)Cite this article

  • 85 Accesses

  • 17 Citations

  • Metrics details

Abstract

We introduce quadri-tilings and show that they are in bijection with dimer models on a family of graphs R * arising from rhombus tilings. Using two height functions, we interpret a sub-family of all quadri-tilings, called triangular quadri-tilings, as an interface model in dimension 2+2. Assigning “critical" weights to edges of R *, we prove an explicit expression, only depending on the local geometry of the graph R *, for the minimal free energy per fundamental domain Gibbs measure; this solves a conjecture of Kenyon (Invent Math 150:409–439, 2002). We also show that when edges of R * are asymptotically far apart, the probability of their occurrence only depends on this set of edges. Finally, we give an expression for a Gibbs measure on the set of all triangular quadri-tilings whose marginals are the above Gibbs measures, and conjecture it to be that of minimal free energy per fundamental domain.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Billingsley P. (1986) Probability Theory. Wiley, New York

    Google Scholar 

  2. Cohn H., Kenyon R., Propp J. (2001) A variational principle for domino tilings. J Am Math Soc 14, 297–346

    Article  MathSciNet  Google Scholar 

  3. Elkies N., Kuperberg G., Larsen M., Propp J. (1992) Alternating sign matrices and domino tilings. J Algebraic Combin 1, 111–132

    Article  MathSciNet  Google Scholar 

  4. Fisher M., Temperley H. (1961) The dimer problem in statistical mechanics—an exact result. Philos Mag 6, 1061–1063

    MathSciNet  Google Scholar 

  5. Kasteleyn P.W. (1961) The statistics of dimers on a lattice. I. The number of dimer arrangements on a quadratic lattice. Physica 27, 1209–1225

    Article  Google Scholar 

  6. Kasteleyn P.W. (1967) Graph theory and crystal physics Graph Theory and Theoretical Physics. Academic Press, London, pp. 43–110

    Google Scholar 

  7. Kenyon R. (1997) Local statistics of lattice dimers. Ann Inst H Poincaré Probab 33, 591–618

    Article  MathSciNet  Google Scholar 

  8. Kenyon R. (2002) The Laplacian and Dirac operators on critical planar graphs. Invent Math 150, 409–439

    Article  MathSciNet  Google Scholar 

  9. Kenyon, R., Okounkov, A., Sheffield, S. Dimers and amoebas, math-ph/0311005. Ann Math (in press) (2006)

  10. Kenyon R., Okounkov, A. Planar dimers and Harnack curves, math.AG/0311062. Duke Math J (in press) (2006)

  11. Kenyon R., Schlenker J.-M. (2005) Rhombic embeddings of planar graphs. Trans Am Math Soc 357(9): 3443–3458

    Article  MathSciNet  Google Scholar 

  12. Kuperberg, G. An exploration of the permanent-determinant method. Electron J Combin 5 (1998), Research Paper 46, 34 pp (electronic)

    Google Scholar 

  13. Mercat C. (2004) Exponentials form a basis of discrete holomorphic functions on a compact. Bull Soc Math Fr 132(2): 305–326

    MathSciNet  Google Scholar 

  14. Mercat, C. Discrete period matrices and related topics, math-ph/0111043 (2001)

  15. Propp, J. Lattice structure for orientations of finite graphs, math.CO 0209005 (1994)

  16. Sheffield, S. PhD Thesis, Stanford University, 2003

  17. Tesler G. (2000) Matchings in graphs on non-orientable surfaces. J Combin Theory Ser B 78(2): 198–231

    Article  MathSciNet  Google Scholar 

  18. Thurston W.P. (1990) Conway’s tiling groups. Am Math Monthly 97, 757–773

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, 8057, Zürich, Switzerland

    Béatrice de Tilière

Authors
  1. Béatrice de Tilière
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Béatrice de Tilière.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

de Tilière, B. Quadri-tilings of the Plane. Probab. Theory Relat. Fields 137, 487–518 (2007). https://doi.org/10.1007/s00440-006-0002-9

Download citation

  • Received: 15 March 2004

  • Revised: 17 February 2006

  • Published: 27 April 2006

  • Issue Date: March 2007

  • DOI: https://doi.org/10.1007/s00440-006-0002-9

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Keywords

  • Dirac Operator
  • Gibbs Measure
  • Minimal Free Energy
  • Height Function
  • Boundary Edge
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature