Skip to Main content Skip to Navigation
Journal articles

Random walks in Weyl chambers and crystals

Abstract : 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.
Complete list of metadata

Cited literature [23 references]  Display  Hide  Download

https://hal.archives-ouvertes.fr/hal-00525592
Contributor : Emmanuel Lesigne Connect in order to contact the contributor
Submitted on : Sunday, December 19, 2010 - 9:53:59 AM
Last modification on : Friday, February 19, 2021 - 4:10:02 PM
Long-term archiving on: : Sunday, March 20, 2011 - 2:25:32 AM

Files

PathCryst_RW.pdf
Files produced by the author(s)

Identifiers

  • HAL Id : hal-00525592, version 2
  • ARXIV : 1010.2341

Collections

Citation

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

Share

Metrics

Record views

427

Files downloads

432