Large deviation principle for the largest eigenvalue of random matrices with a variance profile
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arXiv
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| Format: | Preprint |
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2024
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| author | Ducatez, Raphaël Guionnet, Alice Husson, Jonathan |
| author_facet | Ducatez, Raphaël Guionnet, Alice Husson, Jonathan |
| contents | We establish large deviation principles for the largest eigenvalue of large random matrices with variance profiles. For $N \in \mathbb N$, we consider random $N \times N$ symmetric matrices $H^N$ which are such that $H_{ij}^{N}=\frac{1}{\sqrt{N}}X_{i,j}^{N}$ for $1 \leq i,j \leq N$, where the $X_{i,j}^{N}$ for $1 \leq i \leq j \leq N$ are independent and centered. We then denote $Σ_{i,j} ^N = \text{Var} (X_{i,j}^{N}) ( 1 + \textbf{1}_{ i =j})^{-1}$ the variance profile of $H^N$. Our large deviation principle is then stated under the assumption that the $Σ^N$ converge in a certain sense toward a real continuous function $σ$ of $[0,1]^2$ and that the entries of $H^N$ are sharp sub-Gaussian. Our rate function is expressed in terms of the solution of a Dyson equation involving $σ$. This result is a generalization of a previous work by the third author and is new even in the case of Gaussian entries. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_05413 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Large deviation principle for the largest eigenvalue of random matrices with a variance profile Ducatez, Raphaël Guionnet, Alice Husson, Jonathan Probability 60B20, 60F10 We establish large deviation principles for the largest eigenvalue of large random matrices with variance profiles. For $N \in \mathbb N$, we consider random $N \times N$ symmetric matrices $H^N$ which are such that $H_{ij}^{N}=\frac{1}{\sqrt{N}}X_{i,j}^{N}$ for $1 \leq i,j \leq N$, where the $X_{i,j}^{N}$ for $1 \leq i \leq j \leq N$ are independent and centered. We then denote $Σ_{i,j} ^N = \text{Var} (X_{i,j}^{N}) ( 1 + \textbf{1}_{ i =j})^{-1}$ the variance profile of $H^N$. Our large deviation principle is then stated under the assumption that the $Σ^N$ converge in a certain sense toward a real continuous function $σ$ of $[0,1]^2$ and that the entries of $H^N$ are sharp sub-Gaussian. Our rate function is expressed in terms of the solution of a Dyson equation involving $σ$. This result is a generalization of a previous work by the third author and is new even in the case of Gaussian entries. |
| title | Large deviation principle for the largest eigenvalue of random matrices with a variance profile |
| topic | Probability 60B20, 60F10 |
| url | https://arxiv.org/abs/2403.05413 |