Spectrum of Random $d$-regular Graphs Up to the Edge
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2021
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| _version_ | 1866915133283893248 |
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| author | Huang, Jiaoyang Yau, Horng-Tzer |
| author_facet | Huang, Jiaoyang Yau, Horng-Tzer |
| contents | Consider the normalized adjacency matrices of random $d$-regular graphs on $N$ vertices with fixed degree $d\geq3$. We prove that, with probability $1-N^{-1+{\varepsilon}}$ for any ${\varepsilon} >0$, the following two properties hold as $N \to \infty$ provided that $d\geq3$: (i) The eigenvalues are close to the classical eigenvalue locations given by the Kesten-McKay distribution. In particular, the extremal eigenvalues are concentrated with polynomial error bound in $N$, i.e. $λ_2, |λ_N|\leq 2+N^{-c}$. (ii) All eigenvectors of random $d$-regular graphs are completely delocalized. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2102_00963 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Spectrum of Random $d$-regular Graphs Up to the Edge Huang, Jiaoyang Yau, Horng-Tzer Probability Mathematical Physics Combinatorics 60B20, 05C80 Consider the normalized adjacency matrices of random $d$-regular graphs on $N$ vertices with fixed degree $d\geq3$. We prove that, with probability $1-N^{-1+{\varepsilon}}$ for any ${\varepsilon} >0$, the following two properties hold as $N \to \infty$ provided that $d\geq3$: (i) The eigenvalues are close to the classical eigenvalue locations given by the Kesten-McKay distribution. In particular, the extremal eigenvalues are concentrated with polynomial error bound in $N$, i.e. $λ_2, |λ_N|\leq 2+N^{-c}$. (ii) All eigenvectors of random $d$-regular graphs are completely delocalized. |
| title | Spectrum of Random $d$-regular Graphs Up to the Edge |
| topic | Probability Mathematical Physics Combinatorics 60B20, 05C80 |
| url | https://arxiv.org/abs/2102.00963 |