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.



Tail bounds on eigenvalue gaps for GUE
This is studied in 


*Gérard Ben Arous and Paul Bourgade*, [**Extreme gaps between eigenvalues of random matrices**](http://dx.doi.org/10.1214/11-AOP710), *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.


