The Brown measure of the free multiplicative Brownian motion

Abstract

The free multiplicative Brownian motion $b_t$ is the large-$N$ limit of the Brownian motion on $\mathsf{GL}(N;\mathbb{C}),$ in the sense of $\ast$-distributions. The natural candidate for the large-$N$ limit of the empirical distribution of eigenvalues is thus the Brown measure of $b_t$. In previous work, the second and third authors showed that this Brown measure is supported in the closure of a region ${\Sigma }_t$ that appeared work of Biane. In the present paper, we compute the Brown measure completely. It has a continuous density $W_t$ on ${\bar{\Sigma }}_t,$ which is strictly positive and real analytic on ${\Sigma }_t$. This density has a simple form in polar coordinates: \[ W_{t}(r,θ)=\frac{1}{r^{2}}w_{t}(θ), \] where $w_t$ is an analytic function determined by the geometry of the region ${\Sigma }_t$. We show also that the spectral measure of free unitary Brownian motion $u_t$ is a "shadow" of the Brown measure of $b_t$, precisely mirroring the relationship between Wigner's semicircle law and Ginibre's circular law. We develop several new methods, based on stochastic differential equations and PDE, to prove these results.

arXiv commentAdded references to subsequent works building on these results. Made a notational change, replacing the regularization parameter x with epsilon

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