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 It^o processes $(b{r,s}(t)){t\ge 0}$ in a faithful, tracial $W^\ast$-probability space, and we prove that the full process $(B^N{r,s}(t)){t\ge 0}$ 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 $r=s=\frac12$; we thus resolve the open problem of convergence of the Brownian motion on $\mathbb{GL}_N$ posed by Biane in 1997.