The Annals of Probability

The arctic circle boundary and the Airy process

Kurt Johansson

Full-text: Open access


We prove that the, appropriately rescaled, boundary of the north polar region in the Aztec diamond converges to the Airy process. The proof uses certain determinantal point processes given by the extended Krawtchouk kernel. We also prove a version of Propp’s conjecture concerning the structure of the tiling at the center of the Aztec diamond.

Article information

Ann. Probab., Volume 33, Number 1 (2005), 1-30.

First available in Project Euclid: 11 February 2005

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs 15A52

Airy process determinantal process Dimer model random matrices random tiling


Johansson, Kurt. The arctic circle boundary and the Airy process. Ann. Probab. 33 (2005), no. 1, 1--30. doi:10.1214/009117904000000937.

Export citation


  • Adler, M. and van Moerbeke, P. (2003). A PDE for the joint distributions of the Airy process. Preprint. Available at
  • Baik, J., Kriecherbauer, T., MacLaughlin, K.D.T.-R. and Miller, P. (2003). Uniform asymptotics for polynomials orthogonal with respect to a general class of discrete weights and universality results for associated ensembles: Announcement of results. Internat. Math. Res. Notices 15 821--858.
  • Baryshnikov, Yu. (2001). GUES and QUEUES. Probab. Theory Related Fields 119 256--274.
  • Burton, R. and Pemantle, R. (1993). Local characteristics, entropy and limit theorems for spanning trees and domino tilings via transfer-impedances. Ann. Probab. 21 1329--1371.
  • Cohn, H., Elkies, N. and Propp, J. (1996). Local statistics for random domino tilings of the Aztec diamond. Duke Math. J. 85 117--166.
  • Cohn, H., Kenyon, R. and Propp, J. (2001). A variational principle for domino tilings. J. Amer. Math. Soc. 14 297--346.
  • Cohn, H., Larsen, M. and Propp, J. (1998). The shape of a typical boxed plane partition. New York J. Math. 4 137--165.
  • Elkies, N., Kuperberg, G., Larsen, M. and Propp, J. (1992). Alternating-sign matrices and domino tilings. I. J. Algebraic Combin. 1 111--132.
  • Elkies, N., Kuperberg, G., Larsen, M. and Propp, J. (1992). Alternating-sign matrices and domino tilings. II. J. Algebraic Combin. 1 219--234.
  • Ferrari, P. L. and Spohn, H. (2003). Step fluctuations for a faceted crystal. J. Statist. Phys. 113 1--46.
  • Helfgott, H. (2000). Edge effects on local statistics in lattice dimers: A study of the Aztec diamond (finite case). Preprint. Available at
  • Jockush, W., Propp, J. and Shor, P. (1995). Random domino tilings and the arctic circle theorem. Preprint. Available at
  • Johansson, K. (2000). Shape fluctuations and random matrices. Comm. Math. Phys. 209 437--476.
  • Johansson, K. (2001). Discrete orthogonal polynomial ensembles and the Plancherel measure. Ann. of Math. 153 259--296.
  • Johansson, K. (2002). Non-intersecting paths, random tilings and random matrices. Probab. Theory Related Fields 123 225--280.
  • Johansson, K. (2003). Discrete polynuclear growth and determinantal processes. Comm. Math. Phys. 242 277--329.
  • Kasteleyn, P. W. (1963). Dimer statistics and phase transitions. J. Math. Phys. 4 287--293.
  • Kenyon, R. (1997). Local statistics of lattice dimers. Ann. Inst. H. Poincaré Probab. Statist. 33 591--618.
  • Koekoek, R. and Swartouw, R. F. (1998). The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue. Report 97-17, Dept. Technical Mathematics and Informatics, Delft Univ. Technology. Available at
  • König, W., O'Connell, N. and Roch, S. (2002). Non-colliding random walks, tandem queues and discrete orthogonal polynomial ensembles. Electron. J. Probab. 7 1--24.
  • Nordenstam, E. (2004). Discrete polynuclear growth, the Aztec diamond and the Gaussian unitary ensemble. Master's thesis-MAT-2004-03, KTH, Stockholm.
  • Okounkov, A. and Reshetikhin, N. (2003). Correlation function of Schur process with applications to local geometry with application to local geometry of a random 3-dimensional Young diagram. J. Amer. Math. Soc. 16 581--603.
  • Prähofer, M. and Spohn, H. (2002). Scale invariance of the PNG droplet and the Airy process. J. Statist. Phys. 108 1076--1106.
  • Soshnikov, A. (2000). Determinantal random point fields. Russian Math. Surveys 55 923--975.
  • Stanley, R. P. (1999). Enumerative Combinatorics 2. Cambridge Univ. Press.
  • Stembridge, J. R. (1990). Nonintersecting paths, pfaffians, and plane partitions. Adv. in Math. 83 96--131.
  • Tracy, C. A. and Widom, H. (1994). Level spacing distributions and the Airy kernel. Comm. Math. Phys. 159 151--174.
  • Tracy, C. and Widom, H. (2003). A system of differential equations for the Airy process. Electron. Comm. Probab. 8 93--98.