Title: Matrix Random Walks and the Lima Bean Law
Abstract: How do we simulate Brownian motion? Donsker's functional central limit theorem is the key tool: diffusion rescaled random walks converge to Brownian motion. The step distribution can be chosen very generally, so long as the covariance is right.
How do we simulate Brownian motion on a Lie group? With diffusion rescaled random walks on the group! This can be interpreted in several ways (for example the Wong--Zakai theorem for approximating Brownian motion on manifolds). I will discuss an elementary stochastic calculus approach with ``flat steps'' where strong Lp estimates yield almost sure convergence and generalizes to the free world.
What about eigenvalues? As the dimension increases, convergence of eigenvalues of non-normal random matrices is dicey. I will show that eigenvalues of matrix random walks converge to the Lima Bean Law (arising from Brownian motion on GL(N)) so long as the step distribution is unitarily bi-invariant (as in the "Single Ring Theorem") -- which is even more general than Donsker's theorem allows. This requires fine control on the decay of the pseudospectrum, along with a plethora of complex analytic tools from free probability.
This is joint work with Bruce Driver, Brian Hall, Ching Wei Ho, Yuriy Nemish, and Evangelos "Vaki" Nikitopoulos.