Random walks in Weyl chambers and crystals

TL;DR

This paper connects Kashiwara crystal basis theory with random walks in Weyl chambers, showing that certain conditioned Markov chains can be derived from crystal structures and generalized Pitman transforms.

Contribution

It introduces a novel approach to associate random walks with irreducible representations using crystal basis theory and proves their relation to conditioned Markov chains via a generalized Pitman transform.

Findings

The generalized Pitman transform yields a Markov chain from crystal-based random walks.

For minuscule representations, the Markov chain matches the walk conditioned to stay in the dominant weights cone.

A new renewal theorem for lattice random walks underpins the proofs.

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.