The Brown measure of the free multiplicative Brownian motion

Abstract

AbstractThe free multiplicative Brownian motion $$b_{t}$$ b t is the large-N limit of the Brownian motion on $$\mathsf {GL}(N;\mathbb {C}),$$ GL ( N ; C ) , in the sense of $$*$$ ∗ -distributions. The natural candidate for the large-N limit of the empirical distribution of eigenvalues is thus the Brown measure of $$b_{t}$$ b t . In previous work, the second and third authors showed that this Brown measure is supported in the closure of a region $$\Sigma _{t}$$ Σ t that appeared in the work of Biane. In the present paper, we compute the Brown measure completely. It has a continuous density $$W_{t}$$ W t on $$\overline{\Sigma }_{t},$$ Σ ¯ t , which is strictly positive and real analytic on $$\Sigma _{t}$$ Σ t . This density has a simple form in polar coordinates: $$\begin{aligned} W_{t}(r,\theta )=\frac{1}{r^{2}}w_{t}(\theta ), \end{aligned}$$ W t ( r , θ ) = 1 r 2 w t ( θ ) , where $$w_{t}$$ w t is an analytic function determined by the geometry of the region $$\Sigma _{t}$$ Σ t . We show also that the spectral measure of free unitary Brownian motion $$u_{t}$$ u t is a “shadow” of the Brown measure of $$b_{t}$$ b t , precisely mirroring the relationship between the circular and semicircular laws. We develop several new methods, based on stochastic differential equations and PDE, to prove these results.