A path-transformation for random walks and the Robinson-Schensted correspondence期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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

Authors:	Neil O'Connell

Affiliation:	Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom

Abstract:	The author and Marc Yor recently introduced a path-transformation $G^{(k)}$ with the property that, for 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 ${x: x_1lecdotsle x_k}$ . 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 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.

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

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