Outliers of perturbations of banded Toeplitz matrices
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912081192681472 |
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| author | Bordenave, Charles Chapon, François Capitaine, Mireille |
| author_facet | Bordenave, Charles Chapon, François Capitaine, Mireille |
| contents | Toeplitz matrices form a rich class of possibly non-normal matrices whose asymptotic spectral analysis in high dimension is well-understood. The spectra of these matrices are notoriously highly sensitive to small perturbations. In this work, we analyze the spectrum of a banded Toeplitz matrix perturbed by a random matrix with iid entries of variance $σ_n^2 / n$ in the asymptotic of high dimension and $σ_n$ converging to $σ\geq 0$. Our results complement and provide new proofs on recent progresses in the case $σ= 0$. For any $σ\geq 0$, we show that the point process of outlier eigenvalues is governed by a low-dimensional random analytic matrix field, typically Gaussian, alongside an explicit deterministic matrix that captures the algebraic structure of the resonances responsible for the outlier eigenvalues. On our way, we prove a new functional central limit theorem for trace of polynomials in deterministic and random matrices and present new variations around Szego's strong limit theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_16439 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Outliers of perturbations of banded Toeplitz matrices Bordenave, Charles Chapon, François Capitaine, Mireille Probability Spectral Theory Toeplitz matrices form a rich class of possibly non-normal matrices whose asymptotic spectral analysis in high dimension is well-understood. The spectra of these matrices are notoriously highly sensitive to small perturbations. In this work, we analyze the spectrum of a banded Toeplitz matrix perturbed by a random matrix with iid entries of variance $σ_n^2 / n$ in the asymptotic of high dimension and $σ_n$ converging to $σ\geq 0$. Our results complement and provide new proofs on recent progresses in the case $σ= 0$. For any $σ\geq 0$, we show that the point process of outlier eigenvalues is governed by a low-dimensional random analytic matrix field, typically Gaussian, alongside an explicit deterministic matrix that captures the algebraic structure of the resonances responsible for the outlier eigenvalues. On our way, we prove a new functional central limit theorem for trace of polynomials in deterministic and random matrices and present new variations around Szego's strong limit theorem. |
| title | Outliers of perturbations of banded Toeplitz matrices |
| topic | Probability Spectral Theory |
| url | https://arxiv.org/abs/2410.16439 |