Two CLTs for Sparse Random Matrices
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866912164027039744 |
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| author | Diaconu, Simona |
| author_facet | Diaconu, Simona |
| contents | Let $G=G(n,p_n)$ be a homogeneous Erdös-Rényi graph, and $A$ its adjacency matrix with eigenvalues $λ_1(A) \geq λ_2(A) \geq ... \geq λ_n(A).$ Local laws have been used to show that $lambda_2(A)$ can exhibit fundamentally different behaviors: Tracy-Widom ($p_n \gg n^{-2/3}$), normal ($n^{-7/9} \ll p_n \ll~n^{-2/3}$), and a mix of both ($p_n=cn^{-2/3}$). Additionally, this technique renders the largest eigenvalue $λ_1(A),$ separated from the rest of the spectrum for $p_n \gg n^{-1},$ has Gaussian fluctuations when $p_n \geq n^{-1}(\log{n})^{6+c}$ for some $c>0.$ This paper shows this remains true in the range $Bn^{-1}(\log{n})^4 \leq p_n \leq 1-Bn^{-1}(\log{n})^4$ with $B>0$ universal, the tool behind it being a central limit theorem for the eigenvalue statistics of $A$ that is justified via the method of moments. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_09625 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Two CLTs for Sparse Random Matrices Diaconu, Simona Probability Let $G=G(n,p_n)$ be a homogeneous Erdös-Rényi graph, and $A$ its adjacency matrix with eigenvalues $λ_1(A) \geq λ_2(A) \geq ... \geq λ_n(A).$ Local laws have been used to show that $lambda_2(A)$ can exhibit fundamentally different behaviors: Tracy-Widom ($p_n \gg n^{-2/3}$), normal ($n^{-7/9} \ll p_n \ll~n^{-2/3}$), and a mix of both ($p_n=cn^{-2/3}$). Additionally, this technique renders the largest eigenvalue $λ_1(A),$ separated from the rest of the spectrum for $p_n \gg n^{-1},$ has Gaussian fluctuations when $p_n \geq n^{-1}(\log{n})^{6+c}$ for some $c>0.$ This paper shows this remains true in the range $Bn^{-1}(\log{n})^4 \leq p_n \leq 1-Bn^{-1}(\log{n})^4$ with $B>0$ universal, the tool behind it being a central limit theorem for the eigenvalue statistics of $A$ that is justified via the method of moments. |
| title | Two CLTs for Sparse Random Matrices |
| topic | Probability |
| url | https://arxiv.org/abs/2210.09625 |