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  1. Abstract

    The free multiplicative Brownian motion$$b_{t}$$btis the large-Nlimit of the Brownian motion on$$\mathsf {GL}(N;\mathbb {C}),$$GL(N;C),in the sense of$$*$$-distributions. The natural candidate for the large-Nlimit of the empirical distribution of eigenvalues is thus the Brown measure of$$b_{t}$$bt. In previous work, the second and third authors showed that this Brown measure is supported in the closure of a region$$\Sigma _{t}$$Σtthat appeared in the work of Biane. In the present paper, we compute the Brown measure completely. It has a continuous density$$W_{t}$$Wton$$\overline{\Sigma }_{t},$$Σ¯t,which is strictly positive and real analytic on$$\Sigma _{t}$$Σt. This density has a simple form in polar coordinates:$$\begin{aligned} W_{t}(r,\theta )=\frac{1}{r^{2}}w_{t}(\theta ), \end{aligned}$$Wt(r,θ)=1r2wt(θ),where$$w_{t}$$wtis an analytic function determined by the geometry of the region$$\Sigma _{t}$$Σt. We show also that the spectral measure of free unitary Brownian motion$$u_{t}$$utis a “shadow” of the Brown measure of$$b_{t}$$bt, 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.