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 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)?
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asked Sep 15, 2014 at 16:56
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$\begingroup$ not sure if this is what you're asking, but the "Wigner surmise" for the spacing distribution is derived for a 2x2 matrix and is almost indistinguishable from the large-N limit. $\endgroup$– Carlo BeenakkerCommented Sep 15, 2014 at 18:43
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$\begingroup$ @CarloBeenakker I was not aware of that, but there is still a couple percent difference between the Wigner surmise and the limit distribution, so that if you have reasonably large samples from both, they will look different enough for, say Kolmogorov-Smirnov to be sure that the distributions are different. But maybe that's no longer true for, say $10000x10000$ matrices?! $\endgroup$– Igor RivinCommented Sep 15, 2014 at 19:44
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1 Answer
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On the convergence of the nearest neighbour eigenvalue spacing distribution for orthogonal and symplectic ensembles, 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 $N^{-1/2}$.
answered Sep 15, 2014 at 20:25