Deformed Fréchet law for Wigner and sample covariance matrices with tail in crossover regime
Given A n : = 1 n ( a ij ) an n × n symmetric random matrix, with elements above the diagonal given by i.i.d. random variables having mean zero and unit variance. It is known that when lim x → ∞ x 4 P ( | a ij | > x ) = 0 , then fluctuation of the largest eigenvalue of A n follows a Tracy–Widom d...
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| Format: | Article |
| Language: | English |
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Springer Berlin Heidelberg
2024
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| Online Access: | https://hdl.handle.net/1721.1/157265 |
| _version_ | 1824458145345306624 |
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| author | Han, Yi Han, Yi |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Han, Yi Han, Yi |
| author_sort | Han, Yi |
| collection | MIT |
| description | Given A n : = 1 n ( a ij ) an n × n symmetric random matrix, with elements above the diagonal given by i.i.d. random variables having mean zero and unit variance. It is known that when lim x → ∞ x 4 P ( | a ij | > x ) = 0 , then fluctuation of the largest eigenvalue of A n follows a Tracy–Widom distribution. When the law of a ij is regularly varying with index α ∈ ( 0 , 4 ) , then the largest eigenvalue has a Fréchet distribution. An intermediate regime is recently uncovered in Diaconu (Ann Probab 51(2):774–804, 2023): when lim x → ∞ x 4 P ( | a ij | > x ) = c ∈ ( 0 , ∞ ) , then the law of the largest eigenvalue converges to a deformed Fréchet distribution. In this work we vastly extend the scope where the latter distribution may arise. We show that the same deformed Fréchet distribution arises (1) for sparse Wigner matrices with an average of n Ω ( 1 ) nonzero entries on each row; (2) for periodically banded Wigner matrices with bandwidth p n = n O ( 1 ) ; and more generally for weighted adjacency matrices of any k n -regular graphs with k n = n Ω ( 1 ) . In all these cases, we further prove that the joint distribution of the finitely many largest eigenvalues of A n converge to a deformed Poisson process, and that eigenvectors of the outlying eigenvalues of A n are localized, implying a mobility edge phenomenon at the spectral edge 2 for Wigner matrices. The sparser case with average degree n o ( 1 ) is also explored. Our technique extends to sample covariance matrices, proving for the first time that its largest eigenvalue still follows a deformed Fréchet distribution, assuming the matrix entries satisfy lim x → ∞ x 4 P ( | a ij | > x ) = c ∈ ( 0 , ∞ ) . The proof utilizes a universality result recently established by Brailovskaya and Van Handel (Universality and sharp matrix concentration inequalities, 2022). |
| first_indexed | 2025-02-19T04:21:14Z |
| format | Article |
| id | mit-1721.1/157265 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2025-02-19T04:21:14Z |
| publishDate | 2024 |
| publisher | Springer Berlin Heidelberg |
| record_format | dspace |
| spelling | mit-1721.1/1572652025-01-10T04:48:35Z Deformed Fréchet law for Wigner and sample covariance matrices with tail in crossover regime Han, Yi Han, Yi Massachusetts Institute of Technology. Department of Mathematics Given A n : = 1 n ( a ij ) an n × n symmetric random matrix, with elements above the diagonal given by i.i.d. random variables having mean zero and unit variance. It is known that when lim x → ∞ x 4 P ( | a ij | > x ) = 0 , then fluctuation of the largest eigenvalue of A n follows a Tracy–Widom distribution. When the law of a ij is regularly varying with index α ∈ ( 0 , 4 ) , then the largest eigenvalue has a Fréchet distribution. An intermediate regime is recently uncovered in Diaconu (Ann Probab 51(2):774–804, 2023): when lim x → ∞ x 4 P ( | a ij | > x ) = c ∈ ( 0 , ∞ ) , then the law of the largest eigenvalue converges to a deformed Fréchet distribution. In this work we vastly extend the scope where the latter distribution may arise. We show that the same deformed Fréchet distribution arises (1) for sparse Wigner matrices with an average of n Ω ( 1 ) nonzero entries on each row; (2) for periodically banded Wigner matrices with bandwidth p n = n O ( 1 ) ; and more generally for weighted adjacency matrices of any k n -regular graphs with k n = n Ω ( 1 ) . In all these cases, we further prove that the joint distribution of the finitely many largest eigenvalues of A n converge to a deformed Poisson process, and that eigenvectors of the outlying eigenvalues of A n are localized, implying a mobility edge phenomenon at the spectral edge 2 for Wigner matrices. The sparser case with average degree n o ( 1 ) is also explored. Our technique extends to sample covariance matrices, proving for the first time that its largest eigenvalue still follows a deformed Fréchet distribution, assuming the matrix entries satisfy lim x → ∞ x 4 P ( | a ij | > x ) = c ∈ ( 0 , ∞ ) . The proof utilizes a universality result recently established by Brailovskaya and Van Handel (Universality and sharp matrix concentration inequalities, 2022). 2024-10-11T21:09:17Z 2024-10-11T21:09:17Z 2024-10-04 2024-10-06T03:14:05Z Article http://purl.org/eprint/type/JournalArticle https://hdl.handle.net/1721.1/157265 Han, Y. Deformed Fréchet law for Wigner and sample covariance matrices with tail in crossover regime. Probab. Theory Relat. Fields (2024). PUBLISHER_CC en https://doi.org/10.1007/s00440-024-01329-6 Probability Theory and Related Fields Creative Commons Attribution https://creativecommons.org/licenses/by/4.0/ The Author(s) application/pdf Springer Berlin Heidelberg Springer Berlin Heidelberg |
| spellingShingle | Han, Yi Han, Yi Deformed Fréchet law for Wigner and sample covariance matrices with tail in crossover regime |
| title | Deformed Fréchet law for Wigner and sample covariance matrices with tail in crossover regime |
| title_full | Deformed Fréchet law for Wigner and sample covariance matrices with tail in crossover regime |
| title_fullStr | Deformed Fréchet law for Wigner and sample covariance matrices with tail in crossover regime |
| title_full_unstemmed | Deformed Fréchet law for Wigner and sample covariance matrices with tail in crossover regime |
| title_short | Deformed Fréchet law for Wigner and sample covariance matrices with tail in crossover regime |
| title_sort | deformed frechet law for wigner and sample covariance matrices with tail in crossover regime |
| url | https://hdl.handle.net/1721.1/157265 |
| work_keys_str_mv | AT hanyi deformedfrechetlawforwignerandsamplecovariancematriceswithtailincrossoverregime AT hanyi deformedfrechetlawforwignerandsamplecovariancematriceswithtailincrossoverregime |