Smallest Gaps Between Eigenvalues of Random Matrices With Complex Ginibre, Wishart and Universal Unitary Ensembles
Abstract
In this paper we study the limiting distribution of the $k$ smallest gaps between eigenvalues of three kinds of random matrices -- the Ginibre ensemble, the Wishart ensemble and the universal unitary ensemble. All of them follow a Poissonian ansatz. More precisely, for the Ginibre ensemble we have a global result in which the $k$-th smallest gap has typical length $n^{-3/4}$ with density $x^{4k-1}e^{-x^4}$ after normalization. For the Wishart and the universal unitary ensemble, it has typical length $n^{-4/3}$ and has density $x^{3k-1}e^{-x^3}$ after normalization.
- Publication:
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arXiv e-prints
- Pub Date:
- July 2012
- DOI:
- 10.48550/arXiv.1207.4240
- arXiv:
- arXiv:1207.4240
- Bibcode:
- 2012arXiv1207.4240S
- Keywords:
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- Mathematics - Probability;
- 60B20
- E-Print:
- 31 pages, 1 figure