Lozenge Tilings of Hexagons with Cuts and Asymptotic Fluctuations: a New Universality Class

Article
  • 29 Downloads

Abstract

This paper investigates lozenge tilings of non-convex hexagonal regions and more specifically the asymptotic fluctuations of the tilings within and near the strip formed by opposite cuts in the regions, when the size of the regions tend to infinity, together with the cuts. It leads to a new kernel, which is expected to have universality properties.

Keywords

Lozenge tilings Non-convex polygons Kernels Asymptotics 

Mathematics Subject Classification (2010)

Primary: 60G60 60G65 35Q53 Secondary: 60G10 35Q58 

References

  1. 1.
    Adler, M., Chhita, S., Johansson, K., van Moerbeke, P.: Tacnode GUE-minor processes and double Aztec diamonds. Probab. Theory Relat. Fields 162(1-2), 275–325 (2015)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Adler, M., Johansson, K., van Moerbeke, P.: Double Aztec diamonds and the tacnode process. Adv. Math. 252, 518–571 (2014)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Adler, M., Johansson, K., van Moerbeke, P.: Tilings of non-convex polygons, skew-young tableaux and determinantal processes. Comm. Math. Phys. arXiv:1609.06995 (2018)
  4. 4.
    Adler, M., van Moerbeke, P.: Coupled GUE-minor processes. Int. Math. Res. Not. 21, 10987–11044 (2015). arXiv:1312.3859 MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Beffara, V., Chhita, S., Johansson, K.: Airy point process at the liquid-gas boundary. arXiv:1606.08653
  6. 6.
    Betea, D., Bouttier, J., Nejjar, P., Vuletic, M.: The free boundary schur process and applications. arXiv:1704.05809
  7. 7.
    Borodin, A., Gorin, V., Rains, E.M.: q-distributions on boxed plane partitions. Selecta Math. 16, 731–789 (2010)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Borodin, A., Rains, E.M.: Eynard-mehta theorem, schur process, and their pfaffian analogs. J. Stat. Phys. 121(3–4), 291–317 (2005). arXiv:math-ph/0409059 ADSMathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Borodin, A.: Determinantal Point Processes. The Oxford Handbook of Random Matrix Theory, pp. 231–249. Oxford University Press, Oxford (2011)Google Scholar
  10. 10.
    Borodin, A., Duits, M.: Limits of determinantal processes near a tacnode. Ann. Inst. Henri Poincare (B) 47, 243–258 (2011)ADSMathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Borodin, A., Ferrari, P.L.: Anisotropic growth of random surfaces in 2 + 1 dimensions. Comm. Math. Phys 325, 603–684 (2014)ADSMathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Bufetov, A., Knizel, A.: Asymptotics of random domino tilings of rectangular aztec diamonds. arXiv:1604.01491
  13. 13.
    Chhita, S., Johansson, K.: Domino statistics of the two-periodic Aztec diamond. Adv. Math. 294, 37–149 (2016). arXiv:1606.08653 MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Defosseux, M.: Orbit measures, random matrix theory and interlaced determinantal processes. Ann. Inst. H. Poincar Probab. Statist. 46, 209–249 (2010)ADSMathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Duits, M. On global fluctuations for non-colliding processes, arXiv:1510.08248
  16. 16.
    Duse, E., Johansson, K., Metcalfe, A.: The cusp-airy process. Electron. J. Probab. 21(57). arXiv:1510.02057 (2016)
  17. 17.
    Duse, E., Metcalfe, A.: Asymptotic geometry of discrete interlaced patterns: Part I. Int. J. Math. 26, 1550093 (2015)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Duse, E., Metcalfe, A.: Asymptotic geometry of discrete interlaced patterns: Part II. arXiv:1507.00467
  19. 19.
    Johansson, K.: Non-intersecting paths, random tilings and random matrices. Probab. Theory Relat. Fields 123, 225–280 (2002)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Johansson, K.: Discrete polynuclear growth and determinantal processes. Commun. Math. Phys. 242, 277–329 (2003)ADSMathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Johansson, K.: Discrete orthogonal polynomial ensembles and the Plancherel measure. Ann. Math. 153, 259–296 (2001)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Johansson, K.: The arctic circle boundary and the Airy process. Ann. Probab. 33, 1–30 (2005)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Johansson, K.: Edge Fluctuations and Limit Shapes. Harvard Lectures (2016)Google Scholar
  24. 24.
    Johansson, K., Nordenstam, E.: Eigenvalues of GUE minors. Electron. J. Probab. 11, 1342–1371 (2006)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Gorin, V.E.: Bulk universality for random lozenge tilings near straight boundaries and for tensor products, to appear in communications in mathematical physics. arXiv:1603.02707
  26. 26.
    Gorin, V.E., Petrov, L. Universality of local ststistics for noncolliding random walks. arXiv:1608.3243
  27. 27.
    Gorin, V.E.: Nonintersecting paths and the Hahn orthogonal polynomial ensemble. Funct. Anal. Appl. 42, 180–197 (2008)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Kac, M., Ward, J.C.: A combinatorial solution of the two-dimensional Ising model. Phys. Rev. 88, 1332–1337 (1952)ADSCrossRefMATHGoogle Scholar
  29. 29.
    Kasteleyn, P.W.: The statistics of dimers on a lattice. Physica 27, 1209–1225 (1961)ADSCrossRefMATHGoogle Scholar
  30. 30.
    Kasteleyn, P.W.: Graph theory and crystal physics. Graph theory and theoretical physics, pp. 43–110. Academic Press, London (1967)MATHGoogle Scholar
  31. 31.
    Kaufman, B., Onsager, L.: Crystal statistics. III. Short-range order in a binary ising lattice. Phys. Rev. 76, 1244–1252 (1949)ADSCrossRefMATHGoogle Scholar
  32. 32.
    Kenyon, R.: Local statistics of lattice dimers. Ann. Inst. H. Poincaré, Probabilités 33, 591–618 (1997)ADSMathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Kenyon, R., Okounkov, A.: Limit shapes and the complex Burgers equation. Acta Math. 199(2), 263–302 (2007)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Metcalfe, A.: Universality properties of Gelfand-Tsetlin patterns. Probab. Theory Relat. Fields 155(1-2), 303–346 (2013)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Novak, J.: Lozenge tilings and hurwitz numbers. J. Stat. Phys. 161, 509–517 (2015). arXiv:math/0309074 ADSMathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Okounkov, A., Reshetikhin, N.: The birth of a random matrix. Mosc. Math. J. 6(588), 553–566 (2006)MathSciNetMATHGoogle Scholar
  37. 37.
    Petrov, L.: Asymptotics of uniformly random lozenge tilings of polygons. Gaussian free field. Ann. Probab. 43, 1–43 (2015)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Petrov, L.: Asymptotics of random lozenge tilings via Gelfand-Tsetlin schemes. Probab. Theory Relat. Fields 160(3-4), 429–487 (2014)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  • Mark Adler
    • 1
  • Kurt Johansson
    • 2
  • Pierre van Moerbeke
    • 1
    • 3
  1. 1.Department of MathematicsBrandeis UniversityWalthamUSA
  2. 2.Department of MathematicsKTH Royal Institute of TechnologyStockholmSweden
  3. 3.Department of MathematicsUniversité de LouvainLouvain-la-NeuveBelgium

Personalised recommendations