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BIRS Workshop Lecture Videos

Geometric Matrix Brownian Motion and the Lima Bean Law Kemp, Todd


Geometric matrix Brownian motion is the solution (in $N\times N$ matrices) to the stochastic differential equation $dG_t = G_t dZ_t$, $G_0 = I$, where $Z_t$ is a Ginibre Brownian motion (all independent complex Brownian motion entries). It can also be described as the standard Brownian motion on the Lie group $\mathrm{GL}(N,\mathbb{C})$. For $N>2$, with probability $1$ it is not a normal matrix for any $t>0$. Over the last 5 years, we have made progress in understanding its asymptotic moments and fluctuations, but the non-normality (and lack of explicit symmetry) has made understanding its large-$N$ limit empirical eigenvalue distribution quite challenging. The tools around the circular law are now rich and provide a (log) potential course of action to understand the eigenvalues. There are two sides to this problem in general, both quite difficult: proving that the empirical law of eigenvalues converges (which amounts to certain tightness conditions on singular values), and computing what it converges {\em to}. In the case of the geometric matrix Brownian motion, the question of convergence is still a work in progress; but in recent joint work with Bruce Driver and Brian Hall, we have explicitly calculated the limit empirical eigenvalue distribution. It has an analytic density with a nice polar decomposition, supported on a region that resembles a lima bean for small $t>0$, then folds over and becomes a topological annulus when $t>4$. Our methods blend stochastic analysis, complex analysis, and PDE, and approach the log potential in a new way that we hope will be useful in a wider context.

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