Abstract
We show that bounded changes to the boundary of a lozenge tilings do not affect the local behaviour inside the domain. As a consequence we prove the existence of a local limit in all domains with planar boundary. The proof does not rely on any exact solvability of the model beyond its links with uniform spanning trees.
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Berestycki, N., Laslier, B., Ray, G.: A note on dimers and t-graphs (2016). arXiv:1610.07994
Berestycki, N., Laslier, B., Ray, G.: Universality of fluctutations in the dimer model (2016). arXiv:1603.09740
Boutillier, C., Mkrtchyan, S., Reshetikhin, N., Tingley, P.: Random skew plane partitions with a piecewise periodic back wall. Annales Henri Poincaré 13, 271–296 (2012)
Borodin, A.: Periodic schur process and cylindric partitions. Duke Math. J. 140(3), 391–468 (2007)
Cohn, H., Kenyon, R., Propp, J.: A variational principle for domino tilings. J. AMS 14, 297–346 (2001)
Duse, E., Metcalfe, A.: Asymptotic geometry of discrete interlaced patterns: Part I. Int. J. Math. 26, 1550093 (2015). https://doi.org/10.1142/S0129167X15500937
de Tilière, B., Ferrari, P.: Dimer models and random tilings. In: Boutillier, C., Enriquez, N. (eds.) Panoramas et synthèses, vol. 45 (2015)
Gorin, V.: Bulk universality for random lozenge tilings near straight boundaries and for tensor products. Commun. Math. Phys. 354(1), 317–344 (2017)
Gorin, V., Petrov, L.: Universality of local statistics for noncolliding random walks (2016). arXiv:1608.03243
Johansson, K.: Non-intersecting, simple, symmetric random walks and the extended hahn kernel. Annales de l’institut Fourier 55(6), 2129–2145 (2005)
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.: Local statistics of lattice dimers. Annales de Inst. H. Poincaré. Probabilités et Statistiques 33, 591–618 (1997)
Kenyon, R.: Conformal invariance of domino tiling. Ann. Probab. 28, 759–795 (2000)
Kenyon, R., Okounkov, A., Sheffield, S.: Dimers and amoebae. Ann. Math. 163, 1019–1056 (2006)
Kenyon, R., Sheffield, S.: Dimers, tilings and trees. J. Comb. Theory B 92, 295–317 (2004)
Lawler, G.F.: Intersections of Random Walks. Springer, New York (2012)
Mkrtchyan, S.: Scaling limits of random skew plane partitions with arbitrarily sloped back walls. Comm. Math. Phys. 305, 711–739 (2011)
Okounkov, A., Reshetikhin, N.: Correlation function of schur process with application to local geometry of a random 3-dimensional young diagram. J. Am. Math. Soc. 16, 581–603 (2003)
Petrov, L.: Asymptotics of random lozenge tilings via gelfand-tsetlin schemes. Probab. Theory Relat. Fields 160(3), 429–487 (2014)
Russkikh, M: Dimers in piecewise temperley domains (2016). arXiv:1611.07884
Schramm, Oded: Scaling limits of loop-erased random walks and uniform spanning trees. Isr. J. Math. 118, 221–288 (2000)
Sheffield, S.: Random Surfaces, vol. 304. Société mathématique de France, Asterisque (2005)
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Laslier, B. Local limits of lozenge tilings are stable under bounded boundary height perturbations. Probab. Theory Relat. Fields 173, 1243–1264 (2019). https://doi.org/10.1007/s00440-018-0853-x
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DOI: https://doi.org/10.1007/s00440-018-0853-x