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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

A path-transformation for random walks and the Robinson-Schensted correspondence


Author: Neil O'Connell
Journal: Trans. Amer. Math. Soc. 355 (2003), 3669-3697
MSC (2000): Primary 05E05, 05E10, 15A52, 60B99, 60G50, 60J27, 60J45, 60J65, 60K25, 82C41
Published electronically: May 29, 2003
MathSciNet review: 1990168
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Abstract: The author and Marc Yor recently introduced a path-transformation $G^{(k)}$ with the property that, for $X$ belonging to a certain class of random walks on $\mathbb{Z}_+^k$, the transformed walk $G^{(k)}(X)$has the same law as the original walk conditioned never to exit the Weyl chamber $\{x: x_1\le\cdots\le x_k\}$. In this paper, we show that $G^{(k)}$ is closely related to the Robinson-Schensted algorithm, and use this connection to give a new proof of the above representation theorem. The new proof is valid for a larger class of random walks and yields additional information about the joint law of $X$ and $G^{(k)}(X)$. The corresponding results for the Brownian model are recovered by Donsker's theorem. These are connected with Hermitian Brownian motion and the Gaussian Unitary Ensemble of random matrix theory. The connection we make between the path-transformation $G^{(k)}$ and the Robinson-Schensted algorithm also provides a new formula and interpretation for the latter. This can be used to study properties of the Robinson-Schensted algorithm and, moreover, extends easily to a continuous setting.


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Additional Information

Neil O'Connell
Affiliation: Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom
Email: noc@maths.warwick.ac.uk

DOI: http://dx.doi.org/10.1090/S0002-9947-03-03226-4
PII: S 0002-9947(03)03226-4
Keywords: Pitman's representation theorem, random walk, Brownian motion, Weyl chamber, Young tableau, Robinson-Schensted correspondence, RSK, intertwining, Markov functions, Hermitian Brownian motion, random matrices
Received by editor(s): March 7, 2002
Received by editor(s) in revised form: October 25, 2002
Published electronically: May 29, 2003
Article copyright: © Copyright 2003 American Mathematical Society