Abstract
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.
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References
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)
Berger, N., Biskup, M.: Quenched invariance principle for simple random walk on percolation clusters. Probab. Theory Relat. Fields 137(1–2), 83–120 (2007)
Boutillier, C., de Tilière, B.: Loop statistics in the toroidal honeycomb dimer model. Ann. Probab. 37(5), 1747–1777 (2009)
Chandrasekharan, K.: Elliptic functions. Grundlehren der Mathematischen Wissenschaften, vol. 281. Springer, Berlin (1985)
Chhita, S.: The height fluctuations of an off-critical dimer model on the square grid. J. Stat. Phys. 148(1), 67–88 (2012)
Cimasoni, D., Reshetikhin, N.: Dimers on surface graphs and spin structures I. Comm. Math. Phys. 275(1), 187–208 (2007)
Cimasoni, D., Reshetikhin, N.: Dimers on surface graphs and spin structures II. Comm. Math. Phys. 281(2), 445–468 (2008)
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)
Dubédat, J.: SLE and the free field: partition functions and couplings. J. Am. Math. Soc., 22(4), 995–1054 (2009)
Dubédat, J.: Dimers and analytic torsion I. To appear in J. Am. Math. Soc. (2011) arXiv:1110.2808.
Dubédat, J.: Topics on abelian spin models and related problems. Probab. Surv. 8, 374–402 (2011)
Farkas, H.M., Kra, I.: Graduate Texts in Mathematics. Riemann surfaces, 2nd edn. Springer, New York (1992)
Forman, R.: Determinants of Laplacians on graphs. Topology 32(1), 35–46 (1993)
Kassel, A., Kenyon, R.: Random curves on surfaces induced from the Laplacian determinant. arXiv:1211.6974 (2012)
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)
Kenyon, R.: Conformal invariance of domino tiling. Ann. Probab. 28(2), 759–795 (2000)
Kenyon, R.: Dominos and the Gaussian free field. Ann. Probab. 29(3), 1128–1137 (2001)
Kenyon, R.: The Laplacian and Dirac operators on critical planar graphs. Invent. Math. 150(2), 409–439 (2002)
Kenyon, R.: Lectures on dimers. In Statistical Mechanics, IAS/Park City Math. Ser., vol. 16, pp. 191–230. AMS, Providence (2009)
Kenyon, R.: Spanning forests and the vector bundle Laplacian. Ann. Probab. 39(5), 1983–2017 (2011)
Kenyon, R., Okounkov, A., Sheffield, S.: Dimers and amoebae. Ann. Math. (2), 163(3), 1019–1056 (2006)
Kenyon, R.W., Propp, J.G., Wilson, D.B.: Trees and matchings. Electron. J. Combin., 7, 34 (2000). Research Paper 25.
Kenyon, R.W., Sun, N., Wilson, D.B.: On the asymptotics of dimers on tori. arXiv:1310.2603 (2013)
Lawler, G.F., Limic, V.: Random walk: a modern introduction. Cambridge Studies in Advanced Mathematics, vol. 123. Cambridge University Press, Cambridge (2010)
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)
Lawler, G.F., Werner, W.: The Brownian loop soup. Probab. Theory Relat. Fields 128(4), 565–588 (2004)
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].
Li, Z.: Conformal invariance of isoradial dimers. arXiv:1309.0151 (2013)
Ray, D.B., Singer, I.M.: Analytic torsion for complex manifolds. Ann. Math. 2(98), 154–177 (1973)
Simon, B.: Trace ideals and their applications. Mathematical Surveys and Monographs, 2nd edn., vol. 120. American Mathematical Society, Providence (2005)
Tesler, G.: Matchings in graphs on non-orientable surfaces. J. Combin. Theory Ser. B 78(2), 198–231 (2000)
Thurston, W.P.: Conway’s tiling groups. Amer. Math. Monthly 97(8), 757–773 (1990)
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 York
Yadin, A., Yehudayoff, A.: Loop-erased random walk and Poisson kernel on planar graphs. Ann. Probab. 39(4), 1243–1285 (2011)
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It is our pleasure to thank anonymous referees for their detailed and insightful comments. Partially supported by NSF Grant DMS-1005749.
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Dubédat, J., Gheissari, R. Asymptotics of Height Change on Toroidal Temperleyan Dimer Models. J Stat Phys 159, 75–100 (2015). https://doi.org/10.1007/s10955-014-1181-x
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DOI: https://doi.org/10.1007/s10955-014-1181-x