Fusion coefficients and random walks in alcoves

TL;DR

This paper reveals a connection between fusion coefficients and random walks in alcoves, enabling solutions to the Dirichlet problem and demonstrating convergence to Brownian motion on compact Lie groups.

Contribution

It establishes a novel link between fusion coefficients and random walks in alcoves, and applies this to solve the Dirichlet problem and analyze convergence to Brownian motion.

Findings

Solved the Dirichlet problem for random walks in alcoves.

Proved convergence of certain random walks to Brownian motion.

Established a correspondence between conjugacy class hypergroups and fusion hypergroups.

Abstract

We point out a connection between fusion coefficients and random walks in a fixed level alcove associated to the root system of an affine Lie algebra and use this connection to solve completely the Dirichlet problem on such an alcove for a large class of simple random walks. We establish a correspondence between the hypergroup of conjugacy classes of a compact Lie group and the fusion hypergroup. We prove that a random walk in an alcove, obtained with the help of fusion coefficients, converges, after a proper normalization, towards the radial part of a Brownian motion on a compact Lie group.