Random walks in Weyl chambers and crystals

Cédric Lecouvey (1) , Emmanuel Lesigne (1) , Marc Peigné (1)

1 LMPT - Laboratoire de Mathématiques et Physique Théorique

We use Kashiwara crystal basis theory to associate a random walk W to each irreducible representation V of a simple Lie algebra. This is achieved by endowing the crystal attached to V with a (possibly non uniform) probability distribution compatible with its weight graduation. We then prove that the generalized Pitmann transform defined by Biane, Bougerol and O'Connell for similar random walks with uniform distributions yields yet a Markov chain. When the representation is minuscule, and the associated random walk has a drift in the Weyl chamber, we establish that this Markov chain has the same law as W conditionned to never exit the cone of dominant weights. At the heart of our proof is a quotient version of a renewal theorem that we state in the context of general random walks in a lattice.

Mots clés: Representation theory; Kashiwara crystals; conditionned random walk; renewal theorem.; renewal theorem

Domaines: Théorie des représentations [math.RT]; Probabilités [math.PR]

Cédric Lecouvey, Emmanuel Lesigne, Marc Peigné. Random walks in Weyl chambers and crystals. Proceedings of the London Mathematical Society, 2012, 104 (2), pp.323 - 358. ⟨hal-00525592v2⟩