Journal of Statistical Physics

, Volume 108, Issue 5–6, pp 1071–1106 | Cite as

Scale Invariance of the PNG Droplet and the Airy Process

  • Michael Prähofer
  • Herbert Spohn

Abstract

We establish that the static height fluctuations of a particular growth model, the PNG droplet, converges upon proper rescaling to a limit process, which we call the Airy process A(y). The Airy process is stationary, it has continuous sample paths, its single “time” (fixed y) distribution is the Tracy–Widom distribution of the largest eigenvalue of a GUE random matrix, and the Airy process has a slow decay of correlations as y−2. Roughly the Airy process describes the last line of Dyson's Brownian motion model for random matrices. Our construction uses a multi-layer version of the PNG model, which can be analyzed through fermionic techniques. Specializing our result to a fixed value of y, one reobtains the celebrated result of Baik, Deift, and Johansson on the length of the longest increasing subsequence of a random permutation.

Airy process PNG model longest increasing subsequences free fermion techniques 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

REFERENCES

  1. 1.
    P. Meakin, Fractals, Scaling, and Growth Far From Equilibrium (Cambridge University Press, Cambridge, 1998).Google Scholar
  2. 2.
    M. Prähofer and H. Spohn, Universal distributions for growth processes in 1+1 dimensions and random matrices, Phys. Rev. Lett. 84:4882-4885 (2000).Google Scholar
  3. 3.
    J. Baik, P. Deift, and K. Johansson, On the distribution of the length of the longest increasing subsequence in a random permutation, J. Amer. Math. Soc. 12:1189-1178 (1999).Google Scholar
  4. 4.
    C. A. Tracy and H. Widom, Level-spacing distributions and the Airy kernel, Comm. Math. Phys. 159:151-174 (1994).Google Scholar
  5. 5.
    M. Prähofer and H. Spohn, Statistical self-similarity of one-dimensional growth processes, Physica A 279:342-352 (2000).Google Scholar
  6. 6.
    K. Johansson, Non-intersecting paths, random tilings and random matrices, preprint, arXiv: math.PR/0011250.Google Scholar
  7. 7.
    D. J. Gates and M. Westcott, Stationary states of crystal growth in three dimensions, J. Stat. Phys. 81:681-715 (1995).Google Scholar
  8. 8.
    M. Prähofer and H. Spohn, An exactly solved model of three-dimensional surface growth in the anisotropic KPZ regime, J. Stat. Phys. 88:999-1012 (1997).Google Scholar
  9. 9.
    G. Viennot, Une forme géométrique de la correspondence de Robinson-Schensted, in Combinatoire et Représentation du Groupe Symétrique, D. Foata, ed., Lecture Notes in Mathematics, Vol. 579 (Springer-Verlag, Berlin, 1977), pp. 29-58.Google Scholar
  10. 10.
    H. Helfgott, Edge Effects on Local Statistics in Lattice Dimers: A Study of the Aztec Diamond, B.A. thesis, Brandeis University, 1998. arXiv: math.CO/0007136.Google Scholar
  11. 11.
    M. Reed and B. Simon, Methods of Modern Mathematical Physics Vol. IV: Analysis of Operators (Academic Press, New York, 1978).Google Scholar
  12. 12.
    A. Soshnikov, Determinantal random point fields, Russ. Math. Surv. 55:923-975 (2000). arXiv: math.PR/0002099.Google Scholar
  13. 13.
    H. Spohn, Interacting Brownian particles: A study of Dyson's model, in Hydrodynamic Behavior and Interacting Particle Systems, G. Papanicolaou, ed. (Springer-Verlag, New York, 1987).Google Scholar
  14. 14.
    M. Abramowitz and I. A. Stegun (eds.), Pocketbook of Mathematical Functions (Verlag Harri Deutsch, Thun; Frankfurt/Main, 1984).Google Scholar
  15. 15.
    O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics 2 (Springer-Verlag, New York, 1997).Google Scholar
  16. 16.
    K. Johansson, Discrete orthogonal polynomial ensembles and the Plancherel measure, Ann. Math. 153:259-296 (2001).Google Scholar
  17. 17.
    L. J. Landau, Bessel functions: Monotonicity and bounds, J. London Math. Society 61:197-215 (2000).Google Scholar
  18. 18.
    A. Borodin, A. Okounkov, and G. Olshanski, Asymptotics of Plancherel measures for symmetric groups, J. Amer. Math. Soc. 13:481-515 (2000).Google Scholar
  19. 19.
    P. J. Forrester, The spectrum edge of random matrix ensembles, Nucl. Phys. B 402:709-728 (1993).Google Scholar
  20. 20.
    D. C. Mattis and E. H. Lieb, Exact solution of a many-fermion system and its associated Boson field, J. Math. Phys. 6:304-312 (1965).Google Scholar
  21. 21.
    M. Salmhofer, Renormalization, an Introduction, Text and Monographs in Physics (Springer, Berlin, 1999).Google Scholar
  22. 22.
    J. Baik and E. Rains, Symmetrized random permutations, in Random Matrix Models and Their Applications, P. Bleher and A. Its, eds., MSRI Publications, Vol. 40 (Cambridge University Press, Cambridge, 2001), pp. 1-19.Google Scholar
  23. 23.
    O. Kallenberg, Foundations of Modern Probability (Springer-Verlag, New York, 1997).Google Scholar
  24. 24.
    J. Kerstan, Infinitely Divisible Point Processes (Wiley, New York, 1978).Google Scholar
  25. 25.
    A. Okounkov and N. Reshetikhin, Correlation function of Schur process with application to local geometry of a random 3-dimensional Young diagram, preprint, math.CO/ 0107056.Google Scholar

Copyright information

© Plenum Publishing Corporation 2002

Authors and Affiliations

  • Michael Prähofer
    • 1
  • Herbert Spohn
    • 1
  1. 1.Zentrum Mathematik and Physik DepartmentTU MünchenMünchenGermany

Personalised recommendations