The group $\mathrm{GL}(N)$ of invertible complex matrices is a Lie group; equipped with a left-invariant metric, it acquires a canonical (matrix valued) Brownian motion process. It has a large-$N$ limit known as free multiplicative Brownian motion, which was introduced by Biane in 2001. I proved that it is the large-$N$ limit of the $\mathrm{GL}(N)$ Brownian motion, and together with Guillaume Cébron we studied the (Gaussian) fluctuations of this limit.
The big question is: what is the large-$N$ limit of the empirical eigenvalue distribution of this Brownian motion? Simulations show it is quite complicated, supported on a domain that is not simply connected for $t>4$. The limit is (very likely) the Brown measure of $g_t$, which is a fierce object to compute.
In this talk, I will describe very recent joint work with Brian Hall: we explicitly compute the support set of the Brown measure of $g_t$, and show that it asymptotically contains the eigenvalues of the Brownian motion on $\mathrm{GL}(N)$. In the process, we provide a seemingly new description of the support of the Brown measure of any operator.
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