Journal of Statistical Physics

, Volume 159, Issue 1, pp 75–100 | Cite as

Asymptotics of Height Change on Toroidal Temperleyan Dimer Models



The dimer model is an exactly solvable model of planar statistical mechanics. In its critical phase, various aspects of its scaling limit are known to be described by the Gaussian free field. For periodic graphs, criticality is an algebraic condition on the spectral curve of the model, determined by the edge weights (Kenyon et al. in Ann Math (2) 163(3):1019–1056, 2006); isoradial graphs provide another class of critical dimer models, in which the edge weights are determined by the local geometry. In the present article, we consider another class of graphs: general Temperleyan graphs, i.e. graphs arising in the (generalized) Temperley bijection between spanning trees and dimer models. Building in particular on Forman’s formula and representations of Laplacian determinants in terms of Poisson operators, and under a minimal assumption—viz. that the underlying random walk converges to Brownian motion—we show that the natural topological observable on macroscopic tori converges in law to its universal limit, i.e. the law of the periods of the dimer height function converges to that of the periods of a compactified free field.


Dimers Uniform spanning tree Laplacian determinant Gaussian free field 



It is our pleasure to thank anonymous referees for their detailed and insightful comments. Partially supported by NSF Grant DMS-1005749.


  1. 1.
    Alvarez-Gaumé, L., Bost, J.-B., Moore, G., Nelson, P., Vafa, C.: Bosonization on higher genus Riemann surfaces. Comm. Math. Phys. 112(3), 503–552 (1987)CrossRefADSMATHMathSciNetGoogle Scholar
  2. 2.
    Berger, N., Biskup, M.: Quenched invariance principle for simple random walk on percolation clusters. Probab. Theory Relat. Fields 137(1–2), 83–120 (2007)MATHMathSciNetGoogle Scholar
  3. 3.
    Boutillier, C., de Tilière, B.: Loop statistics in the toroidal honeycomb dimer model. Ann. Probab. 37(5), 1747–1777 (2009)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Chandrasekharan, K.: Elliptic functions. Grundlehren der Mathematischen Wissenschaften, vol. 281. Springer, Berlin (1985)Google Scholar
  5. 5.
    Chhita, S.: The height fluctuations of an off-critical dimer model on the square grid. J. Stat. Phys. 148(1), 67–88 (2012)CrossRefADSMATHMathSciNetGoogle Scholar
  6. 6.
    Cimasoni, D., Reshetikhin, N.: Dimers on surface graphs and spin structures I. Comm. Math. Phys. 275(1), 187–208 (2007)CrossRefADSMATHMathSciNetGoogle Scholar
  7. 7.
    Cimasoni, D., Reshetikhin, N.: Dimers on surface graphs and spin structures II. Comm. Math. Phys. 281(2), 445–468 (2008)CrossRefADSMATHMathSciNetGoogle Scholar
  8. 8.
    de Tilière, B.: Scaling limit of isoradial dimer models and the case of triangular quadri-tilings. Ann. Inst. H. Poincaré Probab. Statist. 43(6), 729–750 (2007)CrossRefMATHGoogle Scholar
  9. 9.
    Dubédat, J.: SLE and the free field: partition functions and couplings. J. Am. Math. Soc., 22(4), 995–1054 (2009)Google Scholar
  10. 10.
    Dubédat, J.: Dimers and analytic torsion I. To appear in J. Am. Math. Soc. (2011) arXiv:1110.2808.
  11. 11.
    Dubédat, J.: Topics on abelian spin models and related problems. Probab. Surv. 8, 374–402 (2011)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Farkas, H.M., Kra, I.: Graduate Texts in Mathematics. Riemann surfaces, 2nd edn. Springer, New York (1992)Google Scholar
  13. 13.
    Forman, R.: Determinants of Laplacians on graphs. Topology 32(1), 35–46 (1993)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Kassel, A., Kenyon, R.: Random curves on surfaces induced from the Laplacian determinant. arXiv:1211.6974 (2012)
  15. 15.
    Kasteleyn, P.W.: The statistics of dimers on a lattice I. The number of dimer arrangements on a quadratic lattice. Physica 27, 1209–1225 (1961)CrossRefADSMATHGoogle Scholar
  16. 16.
    Kenyon, R.: Conformal invariance of domino tiling. Ann. Probab. 28(2), 759–795 (2000)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Kenyon, R.: Dominos and the Gaussian free field. Ann. Probab. 29(3), 1128–1137 (2001)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Kenyon, R.: The Laplacian and Dirac operators on critical planar graphs. Invent. Math. 150(2), 409–439 (2002)CrossRefADSMATHMathSciNetGoogle Scholar
  19. 19.
    Kenyon, R.: Lectures on dimers. In Statistical Mechanics, IAS/Park City Math. Ser., vol. 16, pp. 191–230. AMS, Providence (2009)Google Scholar
  20. 20.
    Kenyon, R.: Spanning forests and the vector bundle Laplacian. Ann. Probab. 39(5), 1983–2017 (2011)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Kenyon, R., Okounkov, A., Sheffield, S.: Dimers and amoebae. Ann. Math. (2), 163(3), 1019–1056 (2006)Google Scholar
  22. 22.
    Kenyon, R.W., Propp, J.G., Wilson, D.B.: Trees and matchings. Electron. J. Combin., 7, 34 (2000). Research Paper 25.Google Scholar
  23. 23.
    Kenyon, R.W., Sun, N., Wilson, D.B.: On the asymptotics of dimers on tori. arXiv:1310.2603 (2013)
  24. 24.
    Lawler, G.F., Limic, V.: Random walk: a modern introduction. Cambridge Studies in Advanced Mathematics, vol. 123. Cambridge University Press, Cambridge (2010)Google Scholar
  25. 25.
    Lawler, G.F., Schramm, O., Werner, W.: Conformal invariance of planar loop-erased random walks and uniform spanning trees. Ann. Probab. 32(1B), 939–995 (2004)CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    Lawler, G.F., Werner, W.: The Brownian loop soup. Probab. Theory Relat. Fields 128(4), 565–588 (2004)CrossRefMATHMathSciNetGoogle Scholar
  27. 27.
    Le Jan, Y.: Markov paths, loops and fields. Lecture Notes in Mathematics, vol. 2026. Springer, Heidelberg (2011). Lectures from the 38th Probability Summer School held in Saint-Flour, 2008, École d’Été de Probabilités de Saint-Flour. [Saint-Flour Probability Summer School].Google Scholar
  28. 28.
    Li, Z.: Conformal invariance of isoradial dimers. arXiv:1309.0151 (2013)
  29. 29.
    Ray, D.B., Singer, I.M.: Analytic torsion for complex manifolds. Ann. Math. 2(98), 154–177 (1973)CrossRefMathSciNetGoogle Scholar
  30. 30.
    Simon, B.: Trace ideals and their applications. Mathematical Surveys and Monographs, 2nd edn., vol. 120. American Mathematical Society, Providence (2005)Google Scholar
  31. 31.
    Tesler, G.: Matchings in graphs on non-orientable surfaces. J. Combin. Theory Ser. B 78(2), 198–231 (2000)CrossRefMATHMathSciNetGoogle Scholar
  32. 32.
    Thurston, W.P.: Conway’s tiling groups. Amer. Math. Monthly 97(8), 757–773 (1990)CrossRefMATHMathSciNetGoogle Scholar
  33. 33.
    Wilson, D.B.: Generating random spanning trees more quickly than the cover time. In Proceedings of the Twenty-eighth Annual ACM Symposium on the Theory of Computing (Philadelphia, PA, 1996), pp. 296–303 (1996). ACM, New YorkGoogle Scholar
  34. 34.
    Yadin, A., Yehudayoff, A.: Loop-erased random walk and Poisson kernel on planar graphs. Ann. Probab. 39(4), 1243–1285 (2011)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of MathematicsColumbia UniversityNew YorkUSA
  2. 2.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA

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