Large deviations of the empirical spectral measure of supercritical sparse Wigner matrices
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866914647558324224 |
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| author | Augeri, Fanny |
| author_facet | Augeri, Fanny |
| contents | Let $Ξ$ be the adjacency matrix of an Erdős-Rényi graph on $n$ vertices and with parameter $p$ and consider $A$ a $n\times n$ centered random symmetric matrix with bounded i.i.d. entries above the diagonal. When the mean degree $np$ diverges, the empirical spectral measure of the normalized Hadamard product $(A \circ Ξ)/\sqrt{np}$ converges weakly in probability to the semicircle law. In the regime where $p\ll 1$ and $ np \gg \log n$, we prove a large deviations principle for the empirical spectral measure with speed $n^2p$ and with a good rate function solution of a certain variational problem. The rate function reveals in particular that the only possible deviations at the exponential scale $n^2p$ are around measures coming from Quadratic Vector Equations. As a byproduct, we obtain a large deviations principle for the empirical spectral measure of supercritical Erdős-Rényi graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_11925 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Large deviations of the empirical spectral measure of supercritical sparse Wigner matrices Augeri, Fanny Probability Mathematical Physics Combinatorics Let $Ξ$ be the adjacency matrix of an Erdős-Rényi graph on $n$ vertices and with parameter $p$ and consider $A$ a $n\times n$ centered random symmetric matrix with bounded i.i.d. entries above the diagonal. When the mean degree $np$ diverges, the empirical spectral measure of the normalized Hadamard product $(A \circ Ξ)/\sqrt{np}$ converges weakly in probability to the semicircle law. In the regime where $p\ll 1$ and $ np \gg \log n$, we prove a large deviations principle for the empirical spectral measure with speed $n^2p$ and with a good rate function solution of a certain variational problem. The rate function reveals in particular that the only possible deviations at the exponential scale $n^2p$ are around measures coming from Quadratic Vector Equations. As a byproduct, we obtain a large deviations principle for the empirical spectral measure of supercritical Erdős-Rényi graphs. |
| title | Large deviations of the empirical spectral measure of supercritical sparse Wigner matrices |
| topic | Probability Mathematical Physics Combinatorics |
| url | https://arxiv.org/abs/2401.11925 |