Correlation of a macroscopic dent in a wedge with mixed boundary conditions

Ciucu, Mihai

doi:10.1090/tran/7963

Correlation of a macroscopic dent in a wedge with mixed boundary conditions
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Trans. Amer. Math. Soc. 373 (2020), 2173-2190 Request permission

Abstract:

As part of our ongoing work on the enumeration of symmetry classes of lozenge tilings of hexagons with certain four-lobed structures removed from their center, we consider the case of the tilings which are both vertically and horizontally symmetric. In order to handle this, we need an extension of Kuo’s graphical condensation method, which works in the presence of free boundary. Our results allow us to compute exactly the correlation in a sea of dimers of a macroscopic dent in a 90 degree wedge with mixed boundary conditions. We use previous results to compute the correlation of the corresponding symmetrized system with no boundary and show that its fourth root has the same log-asymptotics as the correlation of the dent in the 90 degree wedge. This is the first result of this kind involving a macroscopic defect. It suggests that the connections between dimer systems with gaps and 2D electrostatics may be deeper than previously thought.

References

George E. Andrews, Plane partitions. I. The MacMahon conjecture, Studies in foundations and combinatorics, Adv. in Math. Suppl. Stud., vol. 1, Academic Press, New York-London, 1978, pp. 131–150. MR 520557
George E. Andrews, Plane partitions. III. The weak Macdonald conjecture, Invent. Math. 53 (1979), no. 3, 193–225. MR 549398, DOI 10.1007/BF01389763
David M. Bressoud, Proofs and confirmations, MAA Spectrum, Mathematical Association of America, Washington, DC; Cambridge University Press, Cambridge, 1999. The story of the alternating sign matrix conjecture. MR 1718370
Mihai Ciucu, A random tiling model for two dimensional electrostatics, Mem. Amer. Math. Soc. 178 (2005), no. 839, x+144. MR 2172582, DOI 10.1090/memo/0839
Mihai Ciucu, The scaling limit of the correlation of holes on the triangular lattice with periodic boundary conditions, Mem. Amer. Math. Soc. 199 (2009), no. 935, x+100. MR 2508012, DOI 10.1090/memo/0935
Mihai Ciucu, The emergence of the electrostatic field as a Feynman sum in random tilings with holes, Trans. Amer. Math. Soc. 362 (2010), no. 9, 4921–4954. MR 2645056, DOI 10.1090/S0002-9947-10-05087-7
Mihai Ciucu, Dimer packings with gaps and electrostatics, Proc. Natl. Acad. Sci. USA 105 (2008), no. 8, 2766–2772. MR 2383565, DOI 10.1073/pnas.0710659105
Mihai Ciucu and Christian Krattenthaler, A dual of MacMahon’s theorem on plane partitions, Proc. Natl. Acad. Sci. USA 110 (2013), no. 12, 4518–4523. MR 3047654, DOI 10.1073/pnas.1217100110
Mihai Ciucu, Lozenge tilings with gaps in a $90^\circ$ wedge domain with mixed boundary conditions, Comm. Math. Phys. 334 (2015), no. 1, 507–532. MR 3304284, DOI 10.1007/s00220-014-2138-2
Mihai Ciucu, Symmetries of shamrocks, Part I, J. Combin. Theory Ser. A 155 (2018), 376–397. MR 3741434, DOI 10.1016/j.jcta.2017.11.013
Mihai Ciucu, Symmetries of shamrocks II: Axial shamrocks, Electron. J. Combin. 25 (2018), no. 2, Paper No. 2.36, 21. MR 3814270
J. W. L. Glaisher, On certain numerical products in which the exponents depend upon the numbers, Messenger Math. 23 (1893), 145–175.
Guy David and Carlos Tomei, The problem of the calissons, Amer. Math. Monthly 96 (1989), no. 5, 429–431. MR 994034, DOI 10.2307/2325150
Christoph Koutschan, Manuel Kauers, and Doron Zeilberger, Proof of George Andrews’s and David Robbins’s $q$-TSPP conjecture, Proc. Natl. Acad. Sci. USA 108 (2011), no. 6, 2196–2199. MR 2775659, DOI 10.1073/pnas.1019186108
C. Krattenthaler, Plane partitions in the work of Richard Stanley and his school, The mathematical legacy of Richard P. Stanley, Amer. Math. Soc., Providence, RI, 2016, pp. 231–261. MR 3618037, DOI 10.1090//mbk/100/14
Eric H. Kuo, Applications of graphical condensation for enumerating matchings and tilings, Theoret. Comput. Sci. 319 (2004), no. 1-3, 29–57. MR 2074946, DOI 10.1016/j.tcs.2004.02.022
E. H. Kuo, Graphical condensation generalizations involving Pfaffians and determinants, preprint arXiv, math:CO/06055154, 2006.
Greg Kuperberg, Symmetries of plane partitions and the permanent-determinant method, J. Combin. Theory Ser. A 68 (1994), no. 1, 115–151. MR 1295786, DOI 10.1016/0097-3165(94)90094-9
Percy A. MacMahon, Combinatory analysis, Chelsea Publishing Co., New York, 1960. Two volumes (bound as one). MR 0141605
Frank W. J. Olver, Asymptotics and special functions, AKP Classics, A K Peters, Ltd., Wellesley, MA, 1997. Reprint of the 1974 original [Academic Press, New York; MR0435697 (55 #8655)]. MR 1429619
Richard P. Stanley, Symmetries of plane partitions, J. Combin. Theory Ser. A 43 (1986), no. 1, 103–113. MR 859302, DOI 10.1016/0097-3165(86)90028-2
John R. Stembridge, Nonintersecting paths, Pfaffians, and plane partitions, Adv. Math. 83 (1990), no. 1, 96–131. MR 1069389, DOI 10.1016/0001-8708(90)90070-4