Skip to main content
Log in

Symmetrized Models of Last Passage Percolation and Non-Intersecting Lattice Paths

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

It has been shown that the last passage time in certain symmetrized models of directed percolation can be written in terms of averages over random matrices from the classical groups U(l), Sp(2l) and O(l). We present a theory of such results based on non-intersecting lattice paths, and integration techniques familiar from the theory of random matrices. Detailed derivations of probabilities relating to two further symmetrizations are also given.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Baik, J., Rains, E.M.: Algebraic aspects of increasing subsequences. Duke Math. J. 109, 1–65 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  2. Baik, J., Rains, E.M.: Symmetrized random permutations. In: Bleher, P.M., Its, A.R. (eds.) Random Matrix Models and Their Applications. Mathematical Sciences Research Institute Publications, vol. 40, pp. 171–208. Cambridge University Press, Cambridge (2001)

    Google Scholar 

  3. Baik, J., Deift, P., Johansson, K.: On the distribution of the length of the longest increasing subsequence of random permutations. J. Am. Math. Soc. 12, 1119–1178 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  4. Borodin, A., Ferrari, P.L., Sasamoto, T.: Transition between Airy1 and Airy2 processes and the TASEP fluctuations. arXiv:math-ph/0703023

  5. Forrester, P.J.: Random walks and random permutations. J. Phys. A 34, L417–L423 (2001)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  6. Forrester, P.J.: Log-gases and random matrices. www.ms.unimelb.edu.au/~matpjf/matpjf.html

  7. Forrester, P.J.: Growth models, random matrices and Painlevé transcendents. Nonlinearity 16, R27–R49 (2003)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  8. Forrester, P.J., Rains, E.M.: Correlations for superpositions and decimations of Laguerre and Jacobi orthogonal matrix ensembles with a parameter. Probab. Theory Relat. Fields 130, 518–576 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  9. Forrester, P.J., Rains, E.M.: Interpretations of some parameter dependent generalizations of classical matrix ensembles. Probab. Theory Relat. Fields 131, 1–61 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  10. Fulton, W.: Young Tableaux. London Mathematical Society Student Texts. Cambridge University Press, Cambridge (1997)

    MATH  Google Scholar 

  11. Gessel, I.M.: Symmetric functions and p-recursiveness. J. Comb. Theory A 53, 257–285 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  12. Gravner, J., Tracy, C.A., Widom, H.: Limit theorems for height functions in a class of discrete space and time growth models. J. Stat. Phys. 102, 1085–1132 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  13. Greene, C.: An extension of Schensted’s theorem. Adv. Math. 14, 254–265 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  14. Johansson, K.: Shape fluctuations and random matrices. Commun. Math. Phys. 209, 437–476 (2000)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  15. Johansson, K.: Non-intersecting paths, random tilings and random matrices. Probab. Theory Relat. Fields 123, 225–280 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  16. Knuth, D.E.: Permutations, matrices and generalized Young tableaux. Pac. J. Math. 34, 709–727 (1970)

    MATH  MathSciNet  Google Scholar 

  17. Krattenthaler, C.: The Major Counting of Nonintersecting Lattice Paths and Generating Functions for Tableaux. Memoirs of the American Mathematical Society, vol. 552. American Mathematical Society, Providence (1995)

    Google Scholar 

  18. Macdonald, I.G.: Hall Polynomials and Symmetric Functions, 2nd edn. Oxford University Press, Oxford (1995)

    Google Scholar 

  19. Mehta, M.L.: Random Matrices, 2nd edn. Academic, New York (1991)

    MATH  Google Scholar 

  20. Rains, E.M.: Increasing subsequences and the classical groups. Electron. J. Comb. 5, #R12 (1998)

    MathSciNet  Google Scholar 

  21. Sagan, B.E.: The Symmetric Group, 2nd edn. Springer, New York (2000)

    Google Scholar 

  22. Spohn, H.: Exact solutions for KPZ-type growth processes, random matrices, and equilibrium shapes for crystals. arXiv:condmat/0512011 (2005)

  23. Stembridge, J.R.: Non-intersecting paths, Pfaffians and plane partitions. Adv. Math. 83, 96–131 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  24. Szegö, G.: Orthogonal Polynomials, 4th edn. American Mathematical Society, Providence (1975)

    MATH  Google Scholar 

  25. van Leeuwen, M.: The Robinson-Schensted and Schützenberger algorithms, an elementary approach. Electron J. Comb. 3, R15 (1996)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Peter J. Forrester.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Forrester, P.J., Rains, E.M. Symmetrized Models of Last Passage Percolation and Non-Intersecting Lattice Paths. J Stat Phys 129, 833–855 (2007). https://doi.org/10.1007/s10955-007-9413-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-007-9413-y

Keywords

Navigation