Extreme gaps between eigenvalues of random matrices

Details Extreme gaps between eigenvalues of random matrices Extreme gaps between eigenvalues of random matrices

2010-10-06

arxiv.org/abs/1010.1294v2

Mathematics

Probability

32 pages, 3 figures

Scientific paper

This paper studies the extreme gaps between eigenvalues of random matrices. We give the joint limiting law of the smallest gaps for Haar-distributed unitary matrices and matrices from the Gaussian Unitary Ensemble. In particular, the $k$th smallest gap, normalized by a factor $n^{-4/3}$, has a limiting density proportional to $x^{3k-1}e^{-x^3}$. Concerning the largest gaps, normalized by $n/\sqrt{\log n}$, they converge in $\L^p$ to a constant for all $p>0$. These results are compared with the extreme gaps between zeros of the Riemann zeta function.

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