In this article, we study the smallest gaps between eigenvalues of the Gaussian orthogonal ensemble (GOE). The main result is that the smallest gaps, after being normalized by n, will converge to a Poisson distribution, and the limiting density of the kth normalized smallest gap is \(2{}x^{2k-1}e^{-x^{2}}/(k-1)!\). The proof is based on the method developed in Feng and Wei (Small gaps of circular \(\beta \)-ensemble. arXiv:1806.01555). We need to prove the convergence of the factorial moments of the smallest gaps, which makes use of the Pfaffian structure of GOE and some comparison results between the one-component log-gas and the two-component log-gas.

中文翻译：

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GOE的小差距

在本文中，我们研究了高斯正交系综（GOE）的特征值之间的最小间隙。主要结果是，最小间隙在被n归一化之后将收敛到泊松分布，第k个归一化最小间隙的极限密度为\（2 {} x ^ {2k-1} e ^ {- x ^ {2}} /（k-1）！\）。证明基于在Feng和Wei中开发的方法（圆\（\ beta \）的小间隙-集合。arXiv：1806.01555）。我们需要证明最小间隙的阶乘矩的收敛性，它利用了GOE的Pfaffian结构以及一元对数气体和二元对数气体之间的一些比较结果。