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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2606.01664 |
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| _version_ | 1866917553519984640 |
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| author | Han, Yi |
| author_facet | Han, Yi |
| contents | Fix a multivariate polynomial $\mathfrak{p}$ in $n$ non-commuting variables of arbitrary degree, and consider $n$ independent $N\times N$ complex Ginibre matrices $X_1^N,\cdots,X_n^N$. We prove that the empirical spectral distribution of $P^N=\mathfrak{p}(X_1^N,\cdots,X_n^N)$ converges as $N$ tends to infinity to the so-called Brown measure of $\mathfrak{p}$ evaluated at free circular variables. For polynomials of degree at most 2, the convergence was proven by Cook, Guionnet, and Husson \cite{cook2022spectrum}, and we prove that the convergence in fact holds for polynomials $\mathfrak{p}$ of any degree. The main step in the proof is a least singular value lower bound for $P^N-z$ for almost all complex shifts $z$, and we prove this via a least singular value lower bound for a wide class of tensorized Ginibre matrices of finite type with a deterministic shift, which is of independent interest. We further show that the Brown measure convergence holds beyond Gaussians: the same convergence holds when the entry law has mean 0, variance 1, bounded density on $\mathbb{C}$ and finite moments of all orders. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2606_01664 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Brown measure convergence for the spectrum of polynomials in Ginibre matrices Han, Yi Probability Fix a multivariate polynomial $\mathfrak{p}$ in $n$ non-commuting variables of arbitrary degree, and consider $n$ independent $N\times N$ complex Ginibre matrices $X_1^N,\cdots,X_n^N$. We prove that the empirical spectral distribution of $P^N=\mathfrak{p}(X_1^N,\cdots,X_n^N)$ converges as $N$ tends to infinity to the so-called Brown measure of $\mathfrak{p}$ evaluated at free circular variables. For polynomials of degree at most 2, the convergence was proven by Cook, Guionnet, and Husson \cite{cook2022spectrum}, and we prove that the convergence in fact holds for polynomials $\mathfrak{p}$ of any degree. The main step in the proof is a least singular value lower bound for $P^N-z$ for almost all complex shifts $z$, and we prove this via a least singular value lower bound for a wide class of tensorized Ginibre matrices of finite type with a deterministic shift, which is of independent interest. We further show that the Brown measure convergence holds beyond Gaussians: the same convergence holds when the entry law has mean 0, variance 1, bounded density on $\mathbb{C}$ and finite moments of all orders. |
| title | Brown measure convergence for the spectrum of polynomials in Ginibre matrices |
| topic | Probability |
| url | https://arxiv.org/abs/2606.01664 |