Probability Theory and Related Fields

, Volume 162, Issue 1–2, pp 275–325 | Cite as

Tacnode GUE-minor processes and double Aztec diamonds

  • Mark Adler
  • Sunil Chhita
  • Kurt Johansson
  • Pierre van Moerbeke


We study determinantal point processes arising in random domino tilings of a double Aztec diamond, a region consisting of two overlapping Aztec diamonds. At a turning point in a single Aztec diamond where the disordered region touches the boundary, the natural limiting process is the GUE-minor process. Increasing the size of a double Aztec diamond while keeping the overlap between the two Aztec diamonds finite, we obtain a new determinantal point process which we call the tacnode GUE-minor process. This process can be thought of as two colliding GUE-minor processes. As part of the derivation of the particle kernel whose scaling limit naturally gives the tacnode GUE-minor process, we find the inverse Kasteleyn matrix for the dimer model version of the Double Aztec diamond.


Interlacing Random tiling Kasteleyn Dimer  Airy process Extended kernels Random Hermitian ensembles 

Mathematics Subject Classification (2010)

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


  1. 1.
    Adler, M., Ferrari, P.L., van Moerbeke, P.: Nonintersecting random walks in the neighborhood of a symmetric tacnode. Ann. Prob. 41(4), 2599–2647 (2013)Google Scholar
  2. 2.
    Adler, M., Johansson, K., van Moerbeke, P.: Double Aztec diamonds and the tacnode process. Adv. Math. 252, 518–571 (2014)Google Scholar
  3. 3.
    Adler, M., van Moerbeke, P.: Coupled GUE-minor processes. arXiv:1312.3859 (2013)
  4. 4.
    Borodin, A., Ferrari, P.L.: Anisotropic growth of random surfaces in \(2+1\) dimensions. Commun. Math. Phys. 325(2), 603–684 (2014)Google Scholar
  5. 5.
    Chhita, S., Johansson, K., Young, B.: Asymptotic domino statistics in the Aztec diamond. arXiv:1212.5414 (2012)
  6. 6.
    Cohn, H., Kenyon, R., Propp, J.: A variational principle for domino tilings. J. Am. Math. Soc. 14(2), 297–346 (electronic) (2001)Google Scholar
  7. 7.
    Delvaux, S.: The tacnode kernel: equality of Riemann-Hilbert and Airy resolvent formulas. arXiv:1211.4845 (2012)
  8. 8.
    Elkies, N., Kuperberg, G., Larsen, M., Propp, J.: Alternating-sign matrices and domino tilings I. J. Algebraic Combin. 1(2), 111–132 (1992)Google Scholar
  9. 9.
    Elkies, N., Kuperberg, G., Larsen, M., Propp, J.: Alternating-sign matrices and domino tilings II. J. Algebraic Combin. 1(3), 219–234 (1992)Google Scholar
  10. 10.
    Ferrari, P.L., Spohn, H.: Step fluctuations for a faceted crystal. J. Stat. Phys. 113(1–2), 1–46 (2003)Google Scholar
  11. 11.
    Ferrari, P.L., Vető, B.: Non-colliding Brownian bridges and the asymmetric tacnode process. Electron. J. Probab. 17(44), 17 (2012)Google Scholar
  12. 12.
    Helfgott, H.: Edge effects on local statistcs in lattice dimers: a study of the Aztec diamond (finite case). arXiv:0007:7136 (2000)
  13. 13.
    Jockusch, W., Propp, J., Shor, P.: Random domino tilings and the Arctic Circle Theorem. (1998)
  14. 14.
    Johansson, K.: Non-intersecting paths, random tilings and random matrices. Prob. Theory Relat. Fields 123(2), 225–280 (2002)Google Scholar
  15. 15.
    Johansson, K.: The arctic circle boundary and the Airy process. Ann. Prob. 33(1), 1–30 (2005)Google Scholar
  16. 16.
    Johansson, K.: Non-colliding Brownian motions and the extended tacnode process. Commun. Math. Phys. 319(1), 231–267 (2013)Google Scholar
  17. 17.
    Johansson, K., Nordenstam, E.: Eigenvalues of GUE minors. Electron. J. Prob. 11(50), 1342–1371 (2006)Google Scholar
  18. 18.
    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)CrossRefMATHGoogle Scholar
  19. 19.
    Kenyon, R.: Local statistics of lattice dimers. Ann. Inst. H. Poincaré Prob. Stat. 33(5), 591–618 (1997)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Luby, M., Randall, D., Sinclair, A.: Markov chain algorithms for planar lattice structures. SIAM J. Comput. 31(1), 167–192 (2001)Google Scholar
  21. 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 (electronic) (2003)Google Scholar
  22. 22.
    Propp, J.: Generalized domino-shuffling. Theor. Comput. Sci. 303(2–3), 267–301 (2003) (Tilings of the plane)Google Scholar
  23. 23.
    Romik, D.: Arctic circles, domino tilings and square Young tableaux. Ann. Prob. 40(2), 611–647 (2012)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Mark Adler
    • 1
  • Sunil Chhita
    • 2
  • Kurt Johansson
    • 3
  • Pierre van Moerbeke
    • 1
    • 4
  1. 1.Department of MathematicsBrandeis UniversityWalthamUSA
  2. 2.Institute of Applied MathematicsUniversity of BonnBonnGermany
  3. 3.Department of MathematicsRoyal Institute of Technology (KTH)StockholmSweden
  4. 4.Department of MathematicsUniversité de LouvainLouvain-la-NeuveBelgium

Personalised recommendations