Abstract
We present here the main result from [8] and explain how to use Kashiwara crystal basis theory to associate a random walk to each minuscule irreducible representation V of a simple Lie algebra; the generalized Pitman transform defined in [10] for similar random walks with uniform distributions yields yet a Markov chain when the crystal attached to V is endowed with a probability distribution compatible with its weight graduation. The main probabilistic argument in our proof is a quotient version of a renewal theorem that we state in the context of general random walks in a lattice [8]. We present some explicit examples, which can be computed using insertion schemes on tableaux described in [9].
The figures have been drawn with the help of J.R. Licois
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Notes
- 1.
in order to simplify the notations, we will omit the (last) coordinates 0 which appear in λ ∈ P +
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Acknowledgements
The authors thank the referee for his very careful reading and many helpful remarks and comments on a preliminary version of this text. M. Peigné especially acknowledges G. Alsmeyer and M. Löwe for their warm hospitality during the conference Random matrices and iterated random functions they organized in Münster, October 4–7, 2011 and where he presented this work.
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Lecouvey, C., Lesigne, E., Peigné, M. (2013). Conditioned Random Walk in Weyl Chambers and Renewal Theory. In: Alsmeyer, G., Löwe, M. (eds) Random Matrices and Iterated Random Functions. Springer Proceedings in Mathematics & Statistics, vol 53. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38806-4_11
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DOI: https://doi.org/10.1007/978-3-642-38806-4_11
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