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The asymptotics of monotone subsequences of involutions

Rains, Eric M. and Baik, Jinho (2001) The asymptotics of monotone subsequences of involutions. Duke Mathematical Journal, 109 (2). pp. 205-281. ISSN 0012-7094.

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We compute the limiting distributions of the lengths of the longest monotone subsequences of random (signed) involutions with or without conditions on the number of fixed points (and negated points) as the sizes of the involutions tend to infinity. The resulting distributions are, depending on the number of fixed points, (1) the Tracy-Widom distributions for the largest eigenvalues of random GOE, GUE, GSE matrices, (2) the normal distribution, or (3) new classes of distributions which interpolate between pairs of the Tracy-Widom distributions. We also consider the second rows of the corresponding Young diagrams. In each case the convergence of moments is also shown. The proof is based on the algebraic work of J. Baik and E. Rains in [7] which establishes a connection between the statistics of random involutions and a family of orthogonal polynomials, and an asymptotic analysis of the orthogonal polynomials which is obtained by extending the Riemann-Hilbert analysis for the orthogonal polynomials by P. Deift, K. Johansson, and Baik in [3].

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Additional Information:© 2001 Duke University Press. Received 23 February 2000. Revision received 5 February 2001. Baik’s work supported in part by a Sloan Doctoral Dissertation Fellowship during the academic year 1998–1999 as a graduate student at Courant Institute of Mathematical Sciences. We would like to thank Percy Deift for helpful discussions and encouragement, especially for his help in proving Lemma 2.1. We would also like to acknowledge many useful conversations and communications with Peter Forrester, Kurt Johansson, Charles Newman, and HaroldWidom. Special thanks are due the referee who gave us crucial advice, improving the exposition of the paper significantly.
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Alfred P. Sloan FoundationUNSPECIFIED
Issue or Number:2
Classification Code:2000 Mathematics Subject Classification: Primary 60C05; Secondary 45E05, 05A05.
Record Number:CaltechAUTHORS:20171106-152554249
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Official Citation:Baik, Jinho; Rains, Eric M. The asymptotics of monotone subsequences of involutions. Duke Math. J. 109 (2001), no. 2, 205--281. doi:10.1215/S0012-7094-01-10921-6.
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:83005
Deposited By: Tony Diaz
Deposited On:07 Nov 2017 03:14
Last Modified:03 Oct 2019 19:01

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