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

TL;DR

The paper resolves the long-standing question of eigenvalue convergence for Brownian motion on $ ext{GL}(N,C)$ by proving that the empirical eigenvalue distribution converges almost surely to a deterministic Brown measure of the product $b_0 b(t)$, where $b(t)$ is a free multiplicative Brownian motion. The authors develop a sharp small-time affine approximation $B^N(t) o I+W^N(t)$ with a quantitative bound, and combine Hermitization with powerful probabilistic and free-probability tools to pass to the large-$N$ limit. Key ingredients include Schwinger–Dyson equations, strong small-time concentration, rigorous control of small singular values, and a free Wegner estimate, culminating in a full convergence theorem for eigenvalues of invariant Brownian motions on $ ext{GL}(N,C)$. This work completes a conjecture of Biane and deepens the connection between random matrix diffusion, Hermitization, and free probability, providing explicit descriptions of the limit Brown measure in terms of the initial data and the elliptic diffusion parameters. The results have broad implications for the spectral analysis of non-normal large random systems and for the analytic structure of free multiplicative Brownian motion.

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

Figures (6)

• Figure 1: Eigenvalues of $B_0^NB^N(1)$ with $N=2000$ and parameters $(\varrho,\zeta)=(2,0.6+\mathbf{i})$, with two different initial conditions. Also plotted is the boundary of $\mathop{\mathrm{Supp}}\nolimits \mu_{b_0,t}$.
• Figure 2: Eigenvalues of $B^N_{1,0}(t)$ with $N=2000$ and $t=3$ (left) and $t=4$ (right). Also plotted is the boundary $\partial\Sigma(1,t)$, cf. \ref{['eq.Sigma.u']}.
• Figure 3: Eigenvalues of $B_0^NB^N_{1,0}(t)$ with $N=2000$ and $t=\frac{2}{3}$ (left) and $t=0.7$ (right), with initial condition $B_0^N$ a unitary $u_6$ with equal mass eigenvalues at the $6$th roots of unity (highlighted in the figures). Also plotted is the boundary $\partial\Sigma(u_6,t)$, cf. \ref{['eq.Sigma.u']}.
• Figure 4: Eigenvalues of $B_0^NB^N(t,\zeta)$ with $N=2000$. On the left, $B^N_0=I$ and $(t,\zeta)=(3,2-\mathbf{i})$; on the right, $B^N_0$ is unitary with $6$th roots of unity as eigenvalues, and $(t,\zeta)= (2/3,-\mathbf{i}/3)$. Also plotted are the boundaries $\partial\Sigma(b_0,t,\zeta)$, cf. Theorem \ref{['thm.Brown.meas.rhozeta']}.
• Figure 5: Eigenvalues of $B_0^NB^N(t,\zeta)$ with $N=2000$. From top to bottom, $t$ ranges through $0.8,1,1.2$; $\zeta=0$ on the left and $\zeta=0.5$ on the right. In all cases, $B_0^N$ is a normal matrix with empirical eigenvalue distribution $\frac{1}{8}(\delta_1+\delta_{-1}+\delta_3+\delta_{-3})+\frac{1}{4}(\delta_{\mathbf{i}}+\delta_{-\mathbf{i}})$. Also plotted are the boundaries $\partial\Sigma(b_0,t,\zeta)$ from \ref{['eq.Sigma.b0']}.

Theorems & Definitions (69)

• Remark 1.1
• Remark 1.2
• Remark 1.3
• Remark 1.4
• Theorem 1.6
• Theorem 1.7
• Lemma 1.9
• Remark 1.10
• Theorem : Full statement of Theorem \ref{['thm.main']}
• Theorem 1.11: DHKBrown,HoZhong2020Brown