Abstract
Random skew plane partitions of large size distributed according to an appropriately scaled Schur process develop limit shapes. In the present work, we consider the limit of large random skew plane partitions where the inner boundary approaches a piecewise linear curve with non-lattice slopes, describing the limit shape and the local fluctuations in various regions. This analysis is fairly similar to that in Okounkov and Reshetikhin (Commun Math Phys 269:571–609, 2007), but we do find some new behavior. For instance, the boundary of the limit shape is now a single smooth (not algebraic) curve, whereas the boundary in Okounkov and Reshetikhin (Commun Math Phys 269:571–609, 2007) is singular. We also observe the bead process introduced in Boutillier (Ann Probab 37(1):107–142, 2009) appearing in the asymptotics at the top of the limit shape.
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Boutillier C.: The bead model and limit behaviors of dimer models. Ann. Probab. 37(1), 107–142 (2009)
Cerf R., Kenyon R.: The low-temperature expansion of the Wulff crystal in the 3D Ising model. Commun. Math. Phys. 222(1), 147–179 (2001)
Cohn H., Larsen M., Propp J.: The shape of a typical boxed plane partition. N. Y. J. Math. 4, 137–165 (1998)
Gorin V.: Nonintersectiong paths and the Hahn orthogonal polynomial ensemble. Funct. Anal. Appl. 42(3), 180–197 (2008)
Johansson K.: Non-intersecting paths, random tilings and random matrices. Probab. Theory Relat. Fields 123(2), 225–280 (2002)
Kasteleyn P.W.: The statistics of dimers on a lattice. Physica 27, 1209–1225 (1961)
Kenyon Richard: Local statistics of lattice dimers. Ann. Inst. H. Poincaré Probab. Stat. 33(5), 591–618 (1997)
Kenyon R.: Height fluctuations in the honeycomb dimer model. Commun. Math. Phys. 281, 675–709 (2008)
Kenyon R., Okounkov A.: Limit shapes and the complex Burgers equation. Acta Math. 199(2), 263–302 (2007)
Kenyon R., Okounkov A.: Sheffield, Scott Dimers and amoebae. Ann. Math. (2) 163(3), 1019–1056 (2006)
McCoy B., Wu F.: The Two-Dimensional Ising Model. Harvard University Press, Cambridge (1973)
Mkrtchyan, S.: Scaling limits of random skew plane partitions with arbitrarily sloped back walls. Commun. Math. Phys. (2011). doi:10.1007/s00220-011-1277-y.
Nienhuis B., Hilhorst H.J., Blöte H.W.J.: Triangular SOS models and cubic-crystal shapes. J. Phys. A 17(18), 3559–3581 (1984)
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) (electronic)
Okounkov A., Reshetikhin N.: Random skew plane partitions and the Pearcey process. Commun. Math. Phys. 269, 571–609 (2007)
Okounkov A., Reshetikhin N.: The birth of a random matrix. Moscow Math. J. 6(3), 553–566 (2006)
Temperley H.N.V., Fisher M.E.: Dimer problem in statistical mechanics—an exact result. Philos. Mag. 6, 1061–1063 (1961)
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Communicated by Bernard Nienhuis.
N. Reshetikhin and P. Tingley were partially supported by the NSF grant DMS-0354321.
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Boutillier, C., Mkrtchyan, S., Reshetikhin, N. et al. Random Skew Plane Partitions with a Piecewise Periodic Back Wall. Ann. Henri Poincaré 13, 271–296 (2012). https://doi.org/10.1007/s00023-011-0120-5
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DOI: https://doi.org/10.1007/s00023-011-0120-5