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arXiv · 1202.3604

Conditioned one-way simple random walk and representation theory

Cédric Lecouvey·Emmanuel Lesigne·Marc Peigné
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Abstract

We call one-way simple random walk a random walk in the quadrant Z_+^n whose increments belong to the canonical base. In relation with representation theory of Lie algebras and superalgebras, we describe the law of such a random walk conditioned to stay in a closed octant, a semi-open octant or other types of semi-groups. The combinatorial representation theory of these algebras allows us to describe a generalized Pitman transformation which realizes the conditioning on the set of paths of the walk. We pursue here in a direction initiated by O'Connell and his coauthors [13,14,2], and also developed in [12]. Our work relies on crystal bases theory and insertion schemes on tableaux described by Kashiwara and his coauthors in [1] and, very recently, in [5].

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Cédric Lecouvey, Emmanuel Lesigne, Marc Peigné. 2012-02-16. Conditioned one-way simple random walk and representation theory. https://arxiv.org/abs/1202.3604

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