The Large-$N$ Limits of Brownian Motions on ${\mathbb{GL}}_N$

Abstract

We introduce a two-parameter family of diffusion processes $(B_{r,s}^N(t){)}_{t\ge 0}$, $r,s>0$, on the general linear group ${\mathbb{GL}}_N$ that are Brownian motions with respect to certain natural metrics on the group. At the same time, we introduce a two-parameter family of free Itô processes $(b_{r,s}(t){)}_{t\ge 0}$ in a faithful, tracial $W^{\ast }$-probability space, and we prove that the full process $(B_{r,s}^N(t){)}_{t\ge 0}$ converges to $(b_{r,s}(t){)}_{t\ge 0}$ in noncommutative distribution as $N\to \infty$ for each $r,s>0$. The processes $(b_{r,s}(t){)}_{t\ge 0}$ interpolate between the free unitary Brownian motion when $(r,s)=(1,0)$, and the free multiplicative Brownian motion when $r=s=\frac{1}{2}$; we thus resolve the open problem of convergence of the Brownian motion on ${\mathbb{GL}}_N$ posed by Biane in 1997.

References in corpus (2)

• Free Convolution Operators and Free Hall Transform
• Heat Kernel Empirical Laws on ${\mathbb{U}}_N$ and ${\mathbb{GL}}_N$