Edge universality of sparse Erdős-Rényi digraphs
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866915492593139712 |
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| author | He, Yukun |
| author_facet | He, Yukun |
| contents | Let $\mathcal A$ be the adjacency matrix of the Erdős-Rényi directed graph $\mathscr G(N,p)$. We denote the eigenvalues of $\mathcal A$ by $λ_1^{\cal A},...,λ^{\cal A}_N$, and $|λ_1^{\cal A}|=\max_i|λ_i^{\cal A}|$. For $N^{-1+o(1)}\leq p\leq 1/2$, we show that \[ \max_{i=2,3,...,N} \bigg|\frac{λ_i^{\mathcal A}}{\sqrt{Np(1-p)}}\bigg| =1+O(N^{-1/2+o(1)}) \] with very high probability. In addition, we prove that near the unit circle, the local eigenvalue statistics of ${\mathcal A}/\sqrt{Np(1-p)}$ coincide with those of the real Ginibre ensemble. As a by-product, we also show that all non-trivial eigenvectors of $\mathcal A$ are completely delocalized.
For Hermitian random matrices, it is known that the edge statistics are sensitive to the sparsity: in the very sparse regime, one needs to remove many noise random variables (which affect both the mean and the fluctuation) to recover the Tracy-Widom distribution. Our results imply that, compared to their analogues in the Hermitian case, the edge statistics of non-Hermitian sparse random matrices are more robust. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_04723 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Edge universality of sparse Erdős-Rényi digraphs He, Yukun Probability Mathematical Physics 60B20, 05C80 Let $\mathcal A$ be the adjacency matrix of the Erdős-Rényi directed graph $\mathscr G(N,p)$. We denote the eigenvalues of $\mathcal A$ by $λ_1^{\cal A},...,λ^{\cal A}_N$, and $|λ_1^{\cal A}|=\max_i|λ_i^{\cal A}|$. For $N^{-1+o(1)}\leq p\leq 1/2$, we show that \[ \max_{i=2,3,...,N} \bigg|\frac{λ_i^{\mathcal A}}{\sqrt{Np(1-p)}}\bigg| =1+O(N^{-1/2+o(1)}) \] with very high probability. In addition, we prove that near the unit circle, the local eigenvalue statistics of ${\mathcal A}/\sqrt{Np(1-p)}$ coincide with those of the real Ginibre ensemble. As a by-product, we also show that all non-trivial eigenvectors of $\mathcal A$ are completely delocalized. For Hermitian random matrices, it is known that the edge statistics are sensitive to the sparsity: in the very sparse regime, one needs to remove many noise random variables (which affect both the mean and the fluctuation) to recover the Tracy-Widom distribution. Our results imply that, compared to their analogues in the Hermitian case, the edge statistics of non-Hermitian sparse random matrices are more robust. |
| title | Edge universality of sparse Erdős-Rényi digraphs |
| topic | Probability Mathematical Physics 60B20, 05C80 |
| url | https://arxiv.org/abs/2304.04723 |