Journal of Statistical Physics

, Volume 100, Issue 3–4, pp 523–541 | Cite as

Limiting Distributions for a Polynuclear Growth Model with External Sources

  • Jinho Baik
  • Eric M. Rains


The purpose of this paper is to investigate the limiting distribution functions for a polynuclear growth model with two external sources which was considered by Prähofer and Spohn. Depending on the strength of the sources, the limiting distribution functions are either the Tracy–Widom functions of random matrix theory or a new explicit function which has the special property that its mean is zero. Moreover, we obtain transition functions between pairs of the above distribution functions in suitably scaled limits. There are also similar results for a discrete totally asymmetric exclusion process.

PNG ASEP directed polymer random matrix limiting distribution 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    J. Baik, Random vicious walks and random matrices, LANL e-print math. PR/0001022; http://xxx/lanl/gov, Comm. Pure Appl. Math., to appear.Google Scholar
  2. 2.
    J. Baik, P. Deift, and K. Johansson, On the distribution of the length of the longest increasing subsequence of random permutations, J. Amer. Math. Soc. 12(4):1119–1178 (1999).Google Scholar
  3. 3.
    J. Baik and E. M. Rains, Algebraic aspects of increasing subsequences, LANL e-print math. CO/9905083; http://xxx/lanl/gov.Google Scholar
  4. 4.
    J. Baik and E. M. Rains, The asymptotics of monotone subsequences of involutions, LANL e-print math. CO/9905084; http://xxx/lanl/gov.Google Scholar
  5. 5.
    P. Deift, Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach, Courant Lecture Notes in Mathematics, Vol. 3 (CIMS, New York, NY, 1999).Google Scholar
  6. 6.
    P. Deift and X. Zhou, A steepest descent method for oscillatory Riemman-Hilbert problems; asymptotics for the MKdV equation, Ann. of Math. 137:295–368 (1993).Google Scholar
  7. 7.
    P. Deift and X. Zhou, Asymptotics for the Painlevé II equation, Comm. Pure Appl. Math. 48:277–337 (1995).Google Scholar
  8. 8.
    S. Hastings and J. McLeod, A boundary value problem associated with the second Painlevé transcendent and the Korteweg de Vries equation, Arch. Rational Mech. Anal. 73:31–51 (1980).Google Scholar
  9. 9.
    K. Johansson, Shape fluctuations and random matrices, Comm. Math. Phys. 209(2): 437–476 (1999).Google Scholar
  10. 10.
    M. Mehta, Random Matrices (Academic press, San Diago, 1991), 2nd ed.Google Scholar
  11. 11.
    M. Prähofer and H. Spohn, in preparation.Google Scholar
  12. 12.
    E. M. Rains, A mean identity for longest increasing subsequence problems, in preparation.Google Scholar
  13. 13.
    H. Spohn and M. Prähofer, Statistical self-similarity of one-dimensional growth processes, LANL e-print cond-mat/9910273; http:/xxx/lanl/gov.Google Scholar
  14. 14.
    H. Spohn and M. Prähofer, Universal distributions for growth processes in 1+1 dimensions and random matrices, LANL e-print cond-mat/9912264; http://xxx/lanl/gov.Google Scholar
  15. 15.
    C. Tracy and H. Widom, Level-spacing distributions and the Airy kernel, Comm. Math. Phys. 159:151–174 (1994).Google Scholar
  16. 16.
    C. Racy and H. Widom, On orthogonal and symplectic matrix ensembles, Comm. Math. Phys. 177:727–754 (1996).Google Scholar

Copyright information

© Plenum Publishing Corporation 2000

Authors and Affiliations

  • Jinho Baik
    • 1
    • 2
  • Eric M. Rains
    • 3
  1. 1.Department of MathematicsPrinceton UniversityPrinceton
  2. 2.Institute for Advanced StudyPrinceton
  3. 3.AT&T Research, New JerseyFlorham Park

Personalised recommendations