We construct a sequence of Markov processes on the set of dominant weights of an affine Lie algebra $mathfrak{g}$ considering tensor product of irreducible highest weight modules of $mathfrak{g}$ and specializations of the characters involving the We
yl vector $rho$. We show that it converges towards a space-time Brownian motion with a drift, conditioned to remain in a Weyl chamber associated to the root system of $mathfrak{g}$.
We construct a sequence of Markov processes on the set of dominant weights of the Affine Lie algebra $hat{mathfrak{sl}_2}(C)$ which involves tensor product of irreducible highest weight modules of $hat{mathfrak{sl}_2}(C)$ and show that it converges t
owards a Doobs space-time harmonic transformation of a space-time Brownian motion.
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 clas
s 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.
We have introduced recently a particles model with blocking and pushing interactions which is related to a Pieri type formula for the orthogonal group. This model has a symplectic version presented here.
We introduce a new interacting particles model with blocking and pushing interactions. Particles evolve on the positive line jumping on their own volition rightwards or leftwards according to geometric jumps with parameter q. We show that the model i
nvolves a Pieri-type formula for the orthogonal group. We prove that the two extreme cases - q=0 and q=1 - lead respectively to a random tiling model studied by Borodin and Kuan and to a random matrix model.
