1. An explicit formula for free multiplicative Brownian motions via spherical functions
- Author
-
Auer, Martin and Voit, Michael
- Subjects
Mathematics - Probability ,Mathematical Physics ,Mathematics - Classical Analysis and ODEs ,60B20, 60B15, 60F15, 60J65, 60K35, 70F10, 82C22, 43A62, 43A90, 22E46 - Abstract
After some normalization, the logarithms of the ordered singular values of Brownian motions on $GL(N,\mathbb F)$ with $\mathbb F=\mathbb R, \mathbb C$ form Weyl-group invariant Heckman-Opdam processes on $\mathbb R^N$ of type $A_{N-1}$. We use classical elementary formulas for the spherical functions of $GL(N,\mathbb C)/SU(N)$ and the associated Euclidean spaces $H(N,\mathbb C)$ of Hermitian matrices, and show that in the $GL(N,\mathbb C)$-case, these processes can be also interpreted as ordered eigenvalues of Brownian motions on $H(N,\mathbb C)$ with particular drifts. This leads to an explicit description for the free limits for the associated empirical processes for $N\to\infty$ where these limits are independent from the parameter $k$ of the Heckman-Opdam processes. In particular we get new formulas for the distributions of the free multiplicative Browniam motion of Biane. We also show how this approach works for the root systems $B_N, C_N, D_N$.
- Published
- 2024