Consecutive level spacings in the chiral Gaussian unitary ensemble: From the hard and soft edge to the bulk

Abstract

The local spectral statistics of random matrices forms distinct universality classes, strongly depending on the position in the spectrum. Surprisingly, the spacing between consecutive eigenvalues at the spectral edges has received little attention, where the density diverges or vanishes, respectively. This different behaviour is called hard or soft edge. We show that the spacings at the edges are almost indistinguishable from the spacing in the bulk of the spectrum. We present analytical results for consecutive spacings between the kth and (k + 1)st smallest eigenvalues in the chiral Gaussian unitary ensemble, both for finite- and large-n. The result depends on the number of the generic zero..