Eigenvalues of Brownian Motions on $\mathrm{GL}(N,\mathbb{C})$

Abstract

We prove that the empirical law of eigenvalues of Brownian motion on the Lie Group $\mathrm{GL}(N,\mathbb{C})$ converges almost surely to a deterministic probability measure, characterized by a free stochastic differential equation. This fully resolves a conjecture made by Philippe Biane in 1997. Our analysis includes a family $\{B=B_{\ddot{I},\hat{I}\P }:|\hat{I}\P |<\ddot{I}\}$ of nondegenerate diffusion processes on $\mathrm{GL}(N,\mathbb{C})$ whose laws are invariant under unitary conjugation, with initial distributions assumed to be uniformly bounded and invertible. The crux of our analysis is a strong quantitative approximation of Brownian motion $B(t)$ on $\mathrm{GL}(N,\mathbb{C})$ for small $t$ by a single increment $I+W(t)$, where $W=W_{\ddot{I},\hat{I}\P }$ is an elliptic Brownian motion in the Lie algebra $\mathfrak{gl}(N,\mathbb{C})={\mathbb{M}}_N(\mathbb{C})$. Specifically, for any $t\in [0,1]$ and $\hat{I}´>0$, \[ \mathbb{P}\left(\|B(t)-I-W(t)\|\geq δ\right)\leq \left(C t/δ\right)^{N^{2/3}} \] for a constant $C=C_{\ddot{I}}$. Leveraging independence of multiplicative increments of the Brownian motion then allows us to use powerful (anti-)concentration tools for Gaussian matrices to complete the Hermitization procedure for convergence of eigenvalues.