  We study random points on the real line generated by the eigenvalues in
unitary invariant random matrix ensembles or by more general repulsive particle
systems. As the number of points tends to infinity, we prove convergence of the
empirical distribution of nearest neighbor spacings. We extend existing results
for the spacing distribution in two ways. On the one hand, we believe the
empirical distribution to be of more practical relevance than the so far
considered expected distribution. On the other hand, we use the unfolding, a
non-linear rescaling, which transforms the ensemble such that the density of
particles is asymptotically constant. This allows to consider all empirical
spacings, where previous results were restricted to a tiny fraction of the
particles. Moreover, we prove bounds on the rates of convergence. The main
ingredient for the proof, a strong bulk universality result for correlation
functions in the unfolded setting including optimal rates, should be of
independent interest.
