Timeline for Asymptotic results for smallest gap of Gaussian random matrix
Current License: CC BY-SA 4.0
14 events
| when toggle format | what | by | license | comment | |
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| Oct 18, 2022 at 20:12 | vote | accept | Hermi | ||
| Oct 18, 2022 at 3:48 | vote | accept | Hermi | ||
| Oct 18, 2022 at 20:12 | |||||
| Oct 17, 2022 at 16:05 | comment | added | Carlo Beenakker | Tracy-Widom is really irrelevant for your question; if you order them from small to large, only the first few and the last few are governed by Tracy-Widom, they have a mean spacing of order $n^{-2/3}$; this is called the "edge" of the spectrum; the other eigenvalues, so the vast majority of them, have a mean spacing of order $n^{-1}$ and a smallest spacing of order $n^{-3/2}$; this is called the 'bulk" of the spectrum. | |
| Oct 17, 2022 at 15:45 | comment | added | Hermi |
Thanks! But I am not very familiar with concepts like edge of the spectrum and bulk. So you mean $\delta_{\min}=O_(n^{-3/2})$ is much smaller than $\lambda_2-\lambda_1=O_p(n^{-2/3})$? I also add some details for the asymptotic bound for $\lambda_2-\lambda_1$.
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| Oct 17, 2022 at 6:13 | comment | added | Carlo Beenakker | the Tracy-Widom distribution refers to eigenvalues near the edge of the spectrum; the spacing is much larger there than in the bulk; you asked for the smallest spacing; that is in the bulk of the spectrum, not at the edge. | |
| Oct 16, 2022 at 22:04 | comment | added | Hermi | Also, in this paper arxiv.org/abs/1010.1294. In page 6, it says that $n^{2/3}(\lambda_n-2, .., \lambda_{n-k}) \to $a multivariate Tracy–Widom distribution as $n\to \infty$. So it seems that $n^{2/3}(\lambda_2-\lambda_1)=O_p(1)$. I am confused about this result. Does it contradict with $\delta_{\min} n^{3/2}=O_p(1)$? | |
| Oct 16, 2022 at 21:51 | comment | added | Hermi | Can I ask the meaning of this convergence rate of $\delta_{\min}$? Does it mean $\delta_{\min} n^{3/2}=O_p(1)$? | |
| Oct 16, 2022 at 21:20 | comment | added | Hermi | Thanks! If we do not consider the smallest gap. So this result also holds for $\lambda_2-\lambda_1$ and $\lambda_n-\lambda_1$? | |
| Oct 15, 2022 at 17:24 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
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| Oct 15, 2022 at 16:46 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
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| Oct 15, 2022 at 15:34 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
added the formulas for the general case
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| Oct 15, 2022 at 15:24 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
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| Oct 15, 2022 at 9:45 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
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| Oct 15, 2022 at 9:39 | history | answered | Carlo Beenakker | CC BY-SA 4.0 |