Electronic Journal of Probability

The Cusp-Airy process

Erik Duse, Kurt Johansson, and Anthony Metcalfe

Full-text: Open access


At a typical cusp point of the disordered region in a random tiling model we expect to see a determinantal process called the Pearcey process in the appropriate scaling limit. However, in certain situations another limiting point process appears that we call the Cusp-Airy process, which is a kind of two sided extension of the Airy kernel point process. We will study this problem in a class of random lozenge tiling models coming from interlacing particle systems. The situation was briefly studied previously by Okounkov and Reshetikhin under the name cuspidal turning point but their formula is not completely correct.

Article information

Electron. J. Probab., Volume 21 (2016), paper no. 57, 50 pp.

Received: 1 February 2016
Accepted: 7 July 2016
First available in Project Euclid: 9 September 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52)

discrete interlacing systems random tiling process scaling limit new determinantal point process

Creative Commons Attribution 4.0 International License.


Duse, Erik; Johansson, Kurt; Metcalfe, Anthony. The Cusp-Airy process. Electron. J. Probab. 21 (2016), paper no. 57, 50 pp. doi:10.1214/16-EJP2. https://projecteuclid.org/euclid.ejp/1473424498

Export citation


  • [1] M. Adler, J. Delpine and P. van Moerbecke, Dyson’s Nonintersecting Brownian Motions with a Few Outliers. Communications on Pure and Applied Mathematics 62 (2009), no. 3, 334–395
  • [2] J. Baik, Painlevé formulas of the limiting distributions for nonnull complex sample covariance matrices. Duke Math. J. 133 (2006), no. 2, 205–235.
  • [3] J. Baik, G. Ben Arous and S. Péché, Phase transition of the largest eigenvalue for non-null complex sample covariance matrices. Ann. Probab. 33 (2005), no. 5, 1643–1697
  • [4] J. Baik, T. Kriecherbauer, K.D.T-R. McLaughlin and P.D. Miller, Discrete Orthogonal Polynomials: Asymptotics and Applications. Annals of Mathematics Studies Princeton University Press, Princeton, NJ, 2007
  • [5] A. Borodin, V. Gorin and A. Guionnet, Gaussian asymptotics of discrete $\beta $-ensembles. arXiv:1505-03760v1
  • [6] J. Breuer and M. Duits, The Nevai condition and a local law of large numbers for orthogonal polynomial ensembles. Adv. Math. 265 (2014), 441–484
  • [7] P.D. Dragnev and E.B. Saff, Constrained Energy Problems with Applications to Orthogonal Polynomials of a Discrete Variable. J. Anal. Math. 72 (1997), 223–259
  • [8] E. Duse and A. Metcalfe, Asymptotic geometry of discrete interlaced patterns: Part I. arXiv:1412.6653
  • [9] E. Duse and A. Metcalfe, Universal edge fluctuations of discrete interlaced particle systems. (In preparation).
  • [10] E. Duse, Constrained Equilibrium Measure in External Fields and Lozenge Tiling Models. (In preparation).
  • [11] D. Féral, On large deviations for the spectral measure of discrete Coulomb gas. Séminaire de probabilités XLI, 19–49, Lecture Notes in Math. 1934, Springer, Berlin, 2008
  • [12] K. Johansson, Discrete polynuclear growth and determinantal processes. Communications in Mathematical Physics 242 (2003), 277–329.
  • [13] K. Johansson, Random matrices and determinantal processes. Math. Stat. Phy, Session LXXXIII: Lecture Notes of the Les Houches Summer School, Elsevier Science. Elsevier B. V., Amsterdam, 2006
  • [14] A. Metcalfe, Universality properties of Gelfand-Tsetlin patterns. Probability Theory and Related Fields (2013), no. 1-2, 303–346
  • [15] R. Kenyon, Lectures on dimers. Statistical mechanics, 191–230, IAS/Park City Math. Ser., 16, Amer. Math. Soc., Providence, RI, 2009.
  • [16] R. Kenyon and A. Okounkov, What is... a dimer? Notices Amer. Math. Soc. 52 (2005), no. 3, 342–343
  • [17] R. Kenyon and A. Okounkov, Limit shapes and the complex Burgers equation. Acta Mathematica 199 (2007), no. 2, 263–302
  • [18] A. Okounkov and N. Reshetikhin, The birth of a random matrix. Mosc. Math. J. 6 (2006), no. 3, 553–566
  • [19] A. Okounkov and N. Reshetikhin, Random skew plane partitions and the Pearcey Process. Comm. Math. Phys. 269 (2007), no. 3, 571–609
  • [20] L. Petrov, Asymptotics of Random Lozenge Tilings via Gelfand-Tsetlin Schemes. Probab. Theory Related Fields 160 (2014), no. 3-4, 429–487
  • [21] L. Petrov, Asymptotics of uniformly random lozenge tilings of polygons. Gaussian free field. Annals of Probability 43 (2015), no. 1, 1–43.
  • [22] T. Ransford, Potential Theory in the Complex Plane. London Mathematical Society Student Texts, 28 Cambridge University Press, Cambridge, 1995
  • [23] R. Remmert, Classical topics in complex function theory. Graduate Texts in Mathematics 172. Springer-Verlag, New York, 1998
  • [24] H. Weyl, The Classical Groups, Their Invariants and Representations. Princeton University Press, Princeton, NJ, 1997