2025/11/13 by Brailovskaya, Tatiana, Cook, Nicholas A., Kemp, Todd +1
#FOS: Mathematics #Operator Algebras (math.OA) #Probability (math.PR)
paper · doi:10.48550/arxiv.2511.10535
We prove that the empirical law of eigenvalues of Brownian motion on the Lie Group GL(N,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=Bρ,ζ\colon |ζ|<ρ\ of nondegenerate diffusion processes on GL(N,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 GL(N,ℂ) for small t by a single increment I+W(t), where W=Wρ,ζ is an elliptic Brownian motion in the Lie algebra \mathfrakgl(N,ℂ) = \mathbbMN(ℂ). Specifically, for any t∈[0,1] and δ>0, ℙ(‖B(t)-I-W(t)‖≥ δ)≤ (C t/δ)^N2/3 for a constant C=Cρ. 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.