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What I'm looking for is a non-asymptotic bound on the probability that the smallest gap between eigenvalues of a GUE matrix does not exceed a certain value.

I'm aware of the bounds in http://imrn.oxfordjournals.org/content/2010/3/436.full.pdf and https://people.math.osu.edu/nguyen.1261/cikk/gap.pdf , but was wondering if there is a short/elementary way to derive such bounds in the special case of a GUE.

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This is studied in

Gérard Ben Arous and Paul Bourgade, Extreme gaps between eigenvalues of random matrices, Ann. Probab. 41 (2013), no. 4, 2648--2681.

(Ah, so that's how the "insert citation" button works!) In particular, for GUE, the smallest gap has size about $n^{-4/3}$, and after multiplying by $n^{4/3}$, converges to a distribution with pdf $3 x^2 e^{-x^3}$. I think in principle a quantitative rate of convergence can be extracted from the methods of the paper, though it does not appear to have been done explicitly there.

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  • $\begingroup$ Thanks! I was aware of that paper (actually, I was just looking at it). I'll see if I can extract some quantitative bounds from that. $\endgroup$
    – Martin Lotz
    Commented Dec 11, 2015 at 9:09

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