Abstract
In this paper, we introduce a family of observables for the dimer model on a bi-periodic bipartite planar graph, called pattern density fields. We study the scaling limit of these objects for non-frozen Gibbs measures of the dimer model, and prove that they converge to a linear combination of a derivative of the Gaussian massless free field and an independent white noise.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bertini L., Cirillo E.N.M. and Olivieri E. (1999). Renormalization-group transformations under strong mixing conditions: Gibbsianness and convergence of renormalized interactions. J. Stat. Phys. 97: 831–915
Cohn H., Kenyon R. and Propp J. (2001). A variational principle for domino tilings. J.Amer. Math. Soc. 14: 297–346
de Tilière, B.: Conformal invariance of isoradial dimer models & the case of triangular quadri- titlings. http://arxiv.org/list/ math.PR/0512395, 2005
Dobrushin R.L. and Tirozzi B. (1977). The central limit theorem and the problem of equivalence of ensembles. Commun. Math. Phys. 54: 173–192
Fowler R.H. and Rushbrooke G.S. (1937). Statistical theory of perfect solutions. Trans. Faraday Soc. 33: 1272–1294
Giacomin, G., Olla, S., Spohn, H.: Equilibrium fluctuations for \(\nabla\phi\) interface model. Ann. Probab. 29, 1138–1172 (2001)
Glimm J. and Jaffe A. (1981). Quantum physics. Springer-Verlag, New York
Guelfand, I.M., Vilenkin, N.Y.: Les distributions. Tome 4: Applications de l’analyse harmonique, Traduit du russe par G. Rideau. Collection Universitaire de Mathématiques, No. 23, Paris: Dunod, 1967
Iagolnitzer, D., Souillard, B.: Lee-Yang theory and normal fluctuations. Phys. Rev. B (3), 19, 1515–1518 (1979)
Kasteleyn, P.W.: Graph theory and crystal physics. In: Graph Theory and Theoretical Physics, London: Academic Press, 1967, pp. 43–110
Kenyon R. (1997). Local statistics of lattice dimers. Ann. Inst. H. Poincaré Probab. Statist. 33: 591–618
Kenyon R. (2000). Conformal invariance of domino tiling. Ann. Probab. 28: 759–795
Kenyon R. (2001). Dominos and the Gaussian free field. Ann. Probab. 29: 1128–1137
Kenyon R. (2002). The Laplacian and Dirac operators on critical planar graphs. Invent. Math. 150: 409–439
Kenyon, R.: Height fluctuations in the honeycomb dimer model. http://arxiv.org/list/ math-ph/0405052, 2004
Kenyon, R.: An introduction to the dimer model. In: School and Conference on Probability Theory, ICTP Lect. Notes, XVII, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2004, pp. 267–304
Kenyon, R., Okounkov, A.: Limit shapes and the complex Burgers equation. http://arxiv.org/list/math-ph/0507007,2005
Kenyon R., Okounkov A., Sheffield, S.: Dimers and amoebae. Ann. of Math. (2), 163, 1019–1056 (2006)
Naddaf A. and Spencer T. (1997). On homogenization and scaling limit of some gradient perturbations of a massless free field. Comm. Math. Phys. 183: 55–84
Neaderhouser C.C. (1978). Some limit theorems for random fields. Commun. Math. Phys. 61: 293–305
Newman C.M. (1980). Normal fluctuations and the FKG inequalities. Commun. Math. Phys. 74: 119–128
Sheffield, S.: Gaussian Free Field for mathematicians. http://arxiv.org/abs/math/0312099, 2003
Sheffield, S.: Random Surfaces: Large Deviations Principles and Gradient Gibbs Measure Classifications, PhD thesis, Stanford University, 2004, and Asterisque 304, (2005)
Soshnikov A. (2002). Gaussian limit for determinantal random point fields. Ann. Probab. 30: 171–187
Thurston W.P. (1990). Conway’s tiling groups. Amer. Math. Monthly 97: 757–773
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by H. Spohn
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License ( https://creativecommons.org/licenses/by-nc/2.0 ), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Boutillier, C. Pattern Densities in Non-Frozen Planar Dimer Models. Commun. Math. Phys. 271, 55–91 (2007). https://doi.org/10.1007/s00220-006-0175-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-006-0175-1