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Communications in Mathematical Physics

, Volume 252, Issue 1–3, pp 77–109 | Cite as

Polynuclear Growth on a Flat Substrate and Edge Scaling of GOE Eigenvalues

  • Patrik L. FerrariEmail author
Article

Abstract

We consider the polynuclear growth (PNG) model in 1+1 dimension with flat initial condition and no extra constraints. Through the Robinson-Schensted-Knuth (RSK) construction, one obtains the multilayer PNG model, which consists of a stack of non-intersecting lines, the top one being the PNG height. The statistics of the lines is translation invariant and at a fixed position the lines define a point process. We prove that for large times the edge of this point process, suitably scaled, has a limit. This limit is a Pfaffian point process and identical to the one obtained from the edge scaling of the Gaussian orthogonal ensemble (GOE) of random matrices. Our results give further insight to the universality structure within the KPZ class of 1+1 dimensional growth models.

Keywords

Neural Network Complex System Nonlinear Dynamics Growth Model Large Time 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Abramowitz, M., Stegun, I.A.: Pocketbook of Mathematical Functions. Thun-Frankfurt am Main: Verlag Harri Deutsch, 1984Google Scholar
  2. 2.
    Adler, M., van Moerbeke, P.: A PDE for the joint distribution of the Airy process. http:// arxiv.org/abs/math.PR/0302329, 2003, to appear in Ann. Probab.
  3. 3.
    Baik, J., Rains, E.M.: Limiting distributions for a polynuclear growth model with external sources. J. Stat. Phys. 100, 523–542 (2000)CrossRefzbMATHGoogle Scholar
  4. 4.
    Baik, J., Rains, E.M.: Symmetrized random permuations. In: Random Matrix Models and Their Applications, Vol. 40, Cambridge: Cambridge University Press, 2001, pp. 1–19Google Scholar
  5. 5.
    Dyson, F.J.: A Brownian-motion model for the eigenvalues of a random matrix. J. Math. Phys. 3, 1191–1198 (1962)zbMATHGoogle Scholar
  6. 6.
    Forrester, P.J., Nagao, T., Honner, G.: Correlations for the orthogonal-unitary and symplectic-unitary transitions at the hard and soft edges. Nucl Phys. B 553, 601–643 (1999)CrossRefzbMATHGoogle Scholar
  7. 7.
    Gohberg, I.C., Krein, M.G.: Introduction to the theory of nonselfadjoint operators. Transl. Math. Monogr., Vol. 35, Providence, RI: Am. Math. Soc., 1969Google Scholar
  8. 8.
    Greene, C.: An extension of Schensted’s theorem. Adv. Math. 14, 254–265 (1974)zbMATHGoogle Scholar
  9. 9.
    Johansson, K.: The arctic circle boundary and the Airy process. http://arxiv.org/abs/math.PR/ 0306216, 2003, to appear in Ann. Probab. 33 (2005)
  10. 10.
    Karlin, S., McGregor, L.: Coincidence probabilities. Pacific J. 9, 1141–1164 (1959)zbMATHGoogle Scholar
  11. 11.
    Landau, L.J.: Bessel functions: monotonicity and bounds. J. London Math. Soc. 61, 197–215 (2000)CrossRefzbMATHGoogle Scholar
  12. 12.
    Prähofer, M., Spohn, H.: Universal distributions for growth processes in 1+1 dimensions and random matrices. Phys. Rev. Lett. 84, 4882–4885 (2000)CrossRefGoogle Scholar
  13. 13.
    Prähofer, M., Spohn, H.: Current fluctuations for the totally asymmetric simple exclusion process. In: V. Sidoravicius, (ed.), In and out of equilibrium, Progress in Probability, Basel-Boston: Birkhäuser, 2002Google Scholar
  14. 14.
    Prähofer, M., Spohn, H.: Exact scaling function for one-dimensional stationary KPZ growth. J. Stat. Phys. 115, 255–279 (2002)CrossRefGoogle Scholar
  15. 15.
    Prähofer, M., Spohn, H.: Scale invariance of the PNG droplet and the Airy process. J. Stat. Phys. 108, 1071–1106 (2002)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Rains, E.M.: Correlation functions for symmetrized increasing subsequences. http://arxiv.org/abs/math.CO/0006097, 2000
  17. 17.
    Roger, L.C.G., Shi, Z.: Interacting Brownian particles and the Wigner law. Probab. Theory Related Fields 95, 555–570 (1993)Google Scholar
  18. 18.
    Sasamoto, T., Imamura, T.: Fluctuations of a one-dimensional polynuclear growth model in a half space. J. Stat. Phys. 115, 749–803 (2004)CrossRefMathSciNetzbMATHGoogle Scholar
  19. 19.
    Schensted, C.: Longest increasing and decreasing subsequences. Canad. J. Math. 16, 179–191 (1961)Google Scholar
  20. 20.
    Soshnikov, A.: Determinantal random point fields. Russ. Math. Surveys 55, 923–976 (2000)CrossRefzbMATHGoogle Scholar
  21. 21.
    Soshnikov, A.: Janossy densities II. Pfaffian ensembles. J. Stat. Phys. 113, 611–622 (2003)zbMATHGoogle Scholar
  22. 22.
    Stembridge, J.R.: Nonintersecting paths, Pfaffains, and plane partitions. Adv. Math. 83, 96–131 (1990)zbMATHGoogle Scholar
  23. 23.
    Tracy, C.A., Widom, H.: On orthogonal and symplectic matrix ensembles. Commun. Math. Phys. 177, 727–754 (1996)zbMATHGoogle Scholar
  24. 24.
    Tracy, C.A., Widom, H.: Differential equations for Dyson processes. Commun. Math. Phys. 252, 7–41 (2004)Google Scholar
  25. 25.
    Tracy, C.A., Widom, H.: A system of differential equations for the Airy process. Elect. Commun. Probab. 8, 93–98 (2003)Google Scholar
  26. 26.
    Tracy, C.A., Widom, H.: Matrix kernels for the Gaussian orthogonal and symplectic ensembles. http://arxiv.org/abs/math-ph/0405035, 2004
  27. 27.
    Widom, H.: On asymptotic for the Airy process. J. Stat. Phys. 115, 1129–1134 (2004)CrossRefMathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.Zentrum MathematikTechnische Universität MünchenGarching bei MünchenGermany

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