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

TL;DR

This paper proves the almost sure convergence of eigenvalue distributions of Brownian motion on the complex general linear group to a deterministic measure, solving a longstanding conjecture and employing advanced probabilistic and free probability techniques.

Contribution

It fully resolves Philippe Biane's 1997 conjecture by establishing eigenvalue distribution convergence for Brownian motion on ext{GL}(N,C), using novel approximation and concentration methods.

Findings

Eigenvalues converge almost surely to a deterministic measure.

Established a quantitative approximation of Brownian motion for small times.

Applied concentration tools for Gaussian matrices to eigenvalue analysis.

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 $\{B=B_{\rho,\zeta}\colon |\zeta|<\rho\}$ 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 $W=W_{\rho,\zeta}$ 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 $\delta>0$, \[ \mathbb{P}\left(\|B(t)-I-W(t)\|\geq \delta\right)\leq \left(C t/\delta\right)^{N^{2/3}} \] for a constant $C=C_\rho$. 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.