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Conditioned random walks from Kac-Moody root systems


Authors: Cédric Lecouvey, Emmanuel Lesigne and Marc Peigné
Journal: Trans. Amer. Math. Soc. 368 (2016), 3177-3210
MSC (2010): Primary 05E05, 05E10, 60G50, 60J10, 60J22
DOI: https://doi.org/10.1090/tran/6468
Published electronically: July 22, 2015
MathSciNet review: 3451874
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Abstract: Random paths are time continuous interpolations of random
walks. By using the Littelmann path model, we associate to each irreducible highest weight module of a Kac Moody algebra $ {\mathfrak{g}}$ a random path $ {\mathcal {W}}.$ Under suitable hypotheses, we make explicit the probability of the event $ E$: `` $ {\mathcal {W}}$ never exits the Weyl chamber of $ {\mathfrak{g}}$.'' We then give the law of the random walk defined by $ {\mathcal {W}}$ conditioned by the event $ E$ and prove this law can be recovered by applying to $ {\mathcal {W}}$ a path transform of Pitman type. This generalizes the main results of Neil O'Connell (2003) and the authors (2012) to Kac Moody root systems and arbitrary highest weight modules. Our approach here is new and more algebraic than in the aforementioned works. We indeed fully exploit the symmetry of our construction under the action of the Weyl group of $ {\mathfrak{g}}$ which permits us to avoid delicate generalizations of our previous results on renewal theory.


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Additional Information

Cédric Lecouvey
Affiliation: Laboratoire de Mathématiques et Physique Théorique (UMR CNRS 7350), Université François-Rabelais, Tours, Fédération de Recherche Denis Poisson - CNRS, Parc de Grandmont, 37200 Tours, France
Email: cedric.lecouvey@lmpt.univ-tours.fr

Emmanuel Lesigne
Affiliation: Laboratoire de Mathématiques et Physique Théorique (UMR CNRS 7350), Université François-Rabelais, Tours, Fédération de Recherche Denis Poisson - CNRS, Parc de Grandmont, 37200 Tours, France
Email: emmanuel.lesigne@lmpt.univ-tours.fr

Marc Peigné
Affiliation: Laboratoire de Mathématiques et Physique Théorique (UMR CNRS 7350), Université François-Rabelais, Tours, Fédération de Recherche Denis Poisson - CNRS, Parc de Grandmont, 37200 Tours, France
Email: marc.peigne@lmpt.univ-tours.fr

DOI: https://doi.org/10.1090/tran/6468
Received by editor(s): October 18, 2013
Received by editor(s) in revised form: December 21, 2013, and February 21, 2014
Published electronically: July 22, 2015
Article copyright: © Copyright 2015 American Mathematical Society