Spectral gap and edge universality of dense random regular graphs
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866913452897861632 |
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| author | He, Yukun |
| author_facet | He, Yukun |
| contents | Let $\mathcal A$ be the adjacency matrix of a random $d$-regular graph on $N$ vertices, and we denote its eigenvalues by $λ_1\geq λ_2\cdots \geq λ_{N}$. For $N^{2/3}\ll d\leq N/2$, we prove optimal rigidity estimates of the extreme eigenvalues of $\mathcal A$, which in particular imply that \[ \max\{|λ_N|,λ_2\} <2\sqrt{d-1} \] with overwhelming probability. In the same regime of $d$, we also show that \[ N^{2/3}\bigg(\frac{λ_2+d/N}{\sqrt{d(N-d)/N}}-2\bigg) \overset{d}{\longrightarrow} \mathrm{TW}_1\,, \]where $\mathrm{TW}_1$ is the Tracy-Widom distribution for GOE; analogues results also hold for other non-trivial extreme eigenvalues. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2203_07317 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Spectral gap and edge universality of dense random regular graphs He, Yukun Probability 60B20, 15B52, 05C80 Let $\mathcal A$ be the adjacency matrix of a random $d$-regular graph on $N$ vertices, and we denote its eigenvalues by $λ_1\geq λ_2\cdots \geq λ_{N}$. For $N^{2/3}\ll d\leq N/2$, we prove optimal rigidity estimates of the extreme eigenvalues of $\mathcal A$, which in particular imply that \[ \max\{|λ_N|,λ_2\} <2\sqrt{d-1} \] with overwhelming probability. In the same regime of $d$, we also show that \[ N^{2/3}\bigg(\frac{λ_2+d/N}{\sqrt{d(N-d)/N}}-2\bigg) \overset{d}{\longrightarrow} \mathrm{TW}_1\,, \]where $\mathrm{TW}_1$ is the Tracy-Widom distribution for GOE; analogues results also hold for other non-trivial extreme eigenvalues. |
| title | Spectral gap and edge universality of dense random regular graphs |
| topic | Probability 60B20, 15B52, 05C80 |
| url | https://arxiv.org/abs/2203.07317 |