Ramanujan Property and Edge Universality of Random Regular Graphs
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arXiv
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| Natura: | Preprint |
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2024
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| _version_ | 1866917909507342336 |
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| author | Huang, Jiaoyang McKenzie, Theo Yau, Horng-Tzer |
| author_facet | Huang, Jiaoyang McKenzie, Theo Yau, Horng-Tzer |
| contents | We consider the normalized adjacency matrix of a random $d$-regular graph on $N$ vertices with any fixed degree $d\geq 3$ and denote its eigenvalues as $λ_1=d/\sqrt{d-1}\geq λ_2\geqλ_3\cdots\geq λ_N$. We establish the following two results as $N\rightarrow \infty$. (i) With high probability, all eigenvalues are optimally rigid, up to an additional $N^{{\rm o}(1)}$ factor. Specifically, the fluctuations of bulk eigenvalues are bounded by $N^{-1+{\rm o}(1)}$, and the fluctuations of edge eigenvalues are bounded by $N^{-2/3+{\rm o}(1)}$. (ii) Edge universality holds for random $d$-regular graphs. That is, the distributions of $λ_2$ and $-λ_N$ converge to the Tracy-Widom$_1$ distribution associated with the Gaussian Orthogonal Ensemble. As a consequence, for sufficiently large $N$, approximately $69\%$ of $d$-regular graphs on $N$ vertices are Ramanujan, meaning $\max\{λ_2,|λ_N|\}\leq 2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_20263 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Ramanujan Property and Edge Universality of Random Regular Graphs Huang, Jiaoyang McKenzie, Theo Yau, Horng-Tzer Probability Mathematical Physics Combinatorics Spectral Theory 60B20, 05C80, 05C48, 60F05 We consider the normalized adjacency matrix of a random $d$-regular graph on $N$ vertices with any fixed degree $d\geq 3$ and denote its eigenvalues as $λ_1=d/\sqrt{d-1}\geq λ_2\geqλ_3\cdots\geq λ_N$. We establish the following two results as $N\rightarrow \infty$. (i) With high probability, all eigenvalues are optimally rigid, up to an additional $N^{{\rm o}(1)}$ factor. Specifically, the fluctuations of bulk eigenvalues are bounded by $N^{-1+{\rm o}(1)}$, and the fluctuations of edge eigenvalues are bounded by $N^{-2/3+{\rm o}(1)}$. (ii) Edge universality holds for random $d$-regular graphs. That is, the distributions of $λ_2$ and $-λ_N$ converge to the Tracy-Widom$_1$ distribution associated with the Gaussian Orthogonal Ensemble. As a consequence, for sufficiently large $N$, approximately $69\%$ of $d$-regular graphs on $N$ vertices are Ramanujan, meaning $\max\{λ_2,|λ_N|\}\leq 2$. |
| title | Ramanujan Property and Edge Universality of Random Regular Graphs |
| topic | Probability Mathematical Physics Combinatorics Spectral Theory 60B20, 05C80, 05C48, 60F05 |
| url | https://arxiv.org/abs/2412.20263 |