Plane Partitions with Two-Periodic Weights
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We study scaling limits of skew plane partitions with periodic weights under several boundary conditions. We compute the correlation kernel of the limiting point process in the bulk and near turning points on the frozen boundary. The system develops pairs of turning points (points where three different phases meet), which are separated by “semi-frozen” regions. We show that the point process at such a turning point is a pair of non-trivially correlated GUE minor processes. In the limit when all weights become the same, i.e. in the homogeneous case, such a pair of turning points collapses into a single turning point and the local process becomes the GUE minor process. We also study an intermediate regime when the weights are periodic but all converge to 1. In this regime the limit shape and correlations in the bulk are the same as in the case of homogeneous weights, and periodicity is not visible in the bulk. However, the process at turning points is still not the GUE minor process.
Mathematics Subject Classification (2010)Primary 82B20 Secondary 82B05 60C05
KeywordsDimer model plane partitions GUE minor process
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- 1.Betea, D.:Elliptically distributed lozenge tilings of a hexagon (2011). arXiv:1110.4176
- 7.Cohn, H., Larsen, M., Propp, J.: The shape of a typical boxed plane partition. N. Y. J. Math. 4, 137–165 (1998) (electronic)Google Scholar
- 8.Gorin, V.E.: Nonintersecting paths and the Hahn orthogonal polynomial ensemble. Funktsional. Anal. i Prilozhen. 42(3), 23–44, 96 (2008)Google Scholar
- 9.Gorin, V., Panova, G.: Asymptotics of symmetric polynomials with applications to statistical mechanics and representation theory (2013). arXiv:1301.0634
- 18.Okounkov, A.: Symmetric functions and random partitions. In: Symmetric Functions 2001: Surveys of Developments and Perspectives, NATO Sci. Ser. II Math. Phys. Chem., vol. 74, pp. 223–252. Kluwer Acad. Publ., Dordrecht (2002)Google Scholar
- 19.Okounkov, A., Reshetikhin, N.: The birth of a random matrix. Mosc. Math. J. 6(3), 553–566, 588 (2006)Google Scholar
- 21.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(3), 581–603 (2003) (electronic)Google Scholar
- 22.Petrov, L.: Asymptotics of random lozenge tilings via gelfand-tsetlin schemes (2012). arXiv:1202.3901
- 23.Sheffield, S.: Random surfaces. Astérisque (304), vi+175 (2005)Google Scholar