On the distribution of the length of the longest increasing subsequence of random permutations |
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Authors: | Jinho Baik Percy Deift Kurt Johansson |
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Institution: | Courant Institute of Mathematical Sciences, New York University, New York, New York 10012 ; Courant Institute of Mathematical Sciences, New York University, New York, New York 10012 ; Department of Mathematics, Royal Institute of Technology, S-100 44 Stockholm, Sweden |
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Abstract: | The authors consider the length, , of the longest increasing subsequence of a random permutation of numbers. The main result in this paper is a proof that the distribution function for , suitably centered and scaled, converges to the Tracy-Widom distribution of the largest eigenvalue of a random GUE matrix. The authors also prove convergence of moments. The proof is based on the steepest descent method for Riemann-Hilbert problems, introduced by Deift and Zhou in 1993 in the context of integrable systems. The applicability of the Riemann-Hilbert technique depends, in turn, on the determinantal formula of Gessel for the Poissonization of the distribution function of . |
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Keywords: | Random permutations orthogonal polynomials Riemann-Hilbert problems random matrices steepest descent method |
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