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For a symmetric Gaussian random matrix $G=\{G\}_{1\le i,j \le n}$ with iid $E[G_{ij}]=0$ and $E[G_{ij}^2]=1/n$ (it is normalized), ordering its eigenvalues $\lambda_1\le \lambda_2\le\cdots \lambda_n$.

Is there any results about the asymptotic result for the smallest gap $\delta=\min_{1\le i,j \le n}\{|\lambda_i-\lambda_j|\}$?

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For a complex Hermitian matrix (GUE ensemble) the probability distribution of the smallest eigenvalue spacing $\delta_{\rm min}$ is such that the rescaled minimal spacing $x=\delta_{\rm min}n^{4/3}$ has for large $n$ the asymptotic distribution $$P(x)=3x^2e^{-x^3},$$ see Closest Spacing of Eigenvalues by J.P. Vinson (2011).

This result has been generalized to real symmetric matrices (GOE ensemble) by Feng, Tian, and Wei, in Small gaps of GOE (2019). In that case the rescaled minimal spacing $x=\delta_{\rm min}n^{3/2}$ has for large $n$ the limiting distribution $$P(x)=2xe^{-x^2}.$$

Note that for independent eigenvalues (with a mean spacing that scales as $1/n$) the minimal spacing would scale as $n^{-2}$ (Poisson distribution). The $n^{-3/2}$ scaling for the GOE and the $n^{-4/3}$ scaling for the GUE are indicative of "level repulsion" (stronger for the GUE than for the GOE). More generally, for the three classical ensembles (GOE: $\beta=1$, GUE: $\beta=2$, GSE: $\beta=4$) the minimal spacing scales as $n^{-(\beta+2)/(\beta+1)}$, and the rescaled minimal spacing has limiting distribution $P(x)\propto x^\beta e^{-x^{\beta+1}}$.

In the OP the scaling $\delta_{\rm min} n^{2/3}\log n$ is suggested, I don't know where that comes from, it does not agree with these results from the literature, the scaling should be $\delta_{\rm min} n^{3/2}$ in the GOE. Note also that, since the mean level spacing scales as $1/n$, the smallest spacing cannot have an exponent smaller than $1$ -- a scaling as $n^{-2/3}$ is not possible.

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  • $\begingroup$ Thanks! If we do not consider the smallest gap. So this result also holds for $\lambda_2-\lambda_1$ and $\lambda_n-\lambda_1$? $\endgroup$
    – Hermi
    Commented Oct 16, 2022 at 21:20
  • $\begingroup$ Can I ask the meaning of this convergence rate of $\delta_{\min}$? Does it mean $\delta_{\min} n^{3/2}=O_p(1)$? $\endgroup$
    – Hermi
    Commented Oct 16, 2022 at 21:51
  • $\begingroup$ 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 $\delta_{\min} n^{3/2}=O_p(1)$? $\endgroup$
    – Hermi
    Commented Oct 16, 2022 at 22:04
  • $\begingroup$ 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. $\endgroup$
    – Carlo Beenakker
    Commented Oct 17, 2022 at 6:13
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    $\begingroup$ 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. $\endgroup$
    – Carlo Beenakker
    Commented Oct 17, 2022 at 16:05

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