Eigenvalues of Brownian Motions on $\mathrm{GL}(N,\mathbb{C})$
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914156966313984 |
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| author | Brailovskaya, Tatiana Cook, Nicholas A. Kemp, Todd Parraud, Félix |
| author_facet | Brailovskaya, Tatiana Cook, Nicholas A. Kemp, Todd Parraud, Félix |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_10535 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Eigenvalues of Brownian Motions on $\mathrm{GL}(N,\mathbb{C})$ Brailovskaya, Tatiana Cook, Nicholas A. Kemp, Todd Parraud, Félix Probability Operator Algebras 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. |
| title | Eigenvalues of Brownian Motions on $\mathrm{GL}(N,\mathbb{C})$ |
| topic | Probability Operator Algebras |
| url | https://arxiv.org/abs/2511.10535 |