As is well-known (at least in some circles), eigenvalue spacing distribution for large symmetric matrices converges as size goes to infinity (see [this question](https://mathoverflow.net/questions/159987/characterizations-of-the-goe-gue-family-of-distributions) for more background). The question is: how quickly is it known to converge, and is the convergence faster for spacings in the "middle" of the spectrum (presumably there are weird edge effects which muck things up)?



GOE convergence
[On the convergence of the nearest neighbour eigenvalue spacing distribution for orthogonal and symplectic ensembles](http://www-brs.ub.ruhr-uni-bochum.de/netahtml/HSS/Diss/SchubertKristinaBeatrice/diss.pdf), K.B. Schubert (2012).



> 
> In this thesis we consider the empirical distribution of the spacings
>  of adjacent eigenvalues. In particular, we study the expected
>  Kolmogorov-Smirnov distance $D$ of the empirical spacing distribution
>  from its limit as the matrix size $N$ tends to infinity. We find that
>  $D$ converges with rate $N^{-1/2}$.
> 
> 
> 


