Description:

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.