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.
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Baik, J., Rains, E.M.: Algebraic aspects of increasing subsequences. Duke Math. J. 109, 1–65 (2001)
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)
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)
Borodin, A., Ferrari, P.L., Sasamoto, T.: Transition between Airy1 and Airy2 processes and the TASEP fluctuations. arXiv:math-ph/0703023
Forrester, P.J.: Random walks and random permutations. J. Phys. A 34, L417–L423 (2001)
Forrester, P.J.: Log-gases and random matrices. www.ms.unimelb.edu.au/~matpjf/matpjf.html
Forrester, P.J.: Growth models, random matrices and Painlevé transcendents. Nonlinearity 16, R27–R49 (2003)
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)
Forrester, P.J., Rains, E.M.: Interpretations of some parameter dependent generalizations of classical matrix ensembles. Probab. Theory Relat. Fields 131, 1–61 (2005)
Fulton, W.: Young Tableaux. London Mathematical Society Student Texts. Cambridge University Press, Cambridge (1997)
Gessel, I.M.: Symmetric functions and p-recursiveness. J. Comb. Theory A 53, 257–285 (1990)
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)
Greene, C.: An extension of Schensted’s theorem. Adv. Math. 14, 254–265 (1974)
Johansson, K.: Shape fluctuations and random matrices. Commun. Math. Phys. 209, 437–476 (2000)
Johansson, K.: Non-intersecting paths, random tilings and random matrices. Probab. Theory Relat. Fields 123, 225–280 (2002)
Knuth, D.E.: Permutations, matrices and generalized Young tableaux. Pac. J. Math. 34, 709–727 (1970)
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)
Macdonald, I.G.: Hall Polynomials and Symmetric Functions, 2nd edn. Oxford University Press, Oxford (1995)
Mehta, M.L.: Random Matrices, 2nd edn. Academic, New York (1991)
Rains, E.M.: Increasing subsequences and the classical groups. Electron. J. Comb. 5, #R12 (1998)
Sagan, B.E.: The Symmetric Group, 2nd edn. Springer, New York (2000)
Spohn, H.: Exact solutions for KPZ-type growth processes, random matrices, and equilibrium shapes for crystals. arXiv:condmat/0512011 (2005)
Stembridge, J.R.: Non-intersecting paths, Pfaffians and plane partitions. Adv. Math. 83, 96–131 (1990)
Szegö, G.: Orthogonal Polynomials, 4th edn. American Mathematical Society, Providence (1975)
van Leeuwen, M.: The Robinson-Schensted and Schützenberger algorithms, an elementary approach. Electron J. Comb. 3, R15 (1996)
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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
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DOI: https://doi.org/10.1007/s10955-007-9413-y