Abstract
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.
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Acknowledgements
We would like to thank P. Bourgade, O. Zeitouni, G. Ben Arous and P. Forrester for many helpful discussions. We are indebted to the anonymous reviewers for providing many corrections and insightful comments, this paper would not have been possible without their supportive work.
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Feng, R., Tian, G. & Wei, D. Small Gaps of GOE. Geom. Funct. Anal. 29, 1794–1827 (2019). https://doi.org/10.1007/s00039-019-00520-5
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DOI: https://doi.org/10.1007/s00039-019-00520-5