Microscopic and Bulk Spectra of Dirac Operators from Finite-volume Partition Functions

The spectral density ρ(λ) of the Dirac operator in QCD contains interesting informations as it is for example directly proportional to the chiral condensate Σ at the origin through the Banks-Casher relation . While the full spectrum is only accessible numerically in QCD on the lattice many analytic results for parts of the spectrum have been obtained during the past years using random matrix theory (RMT), chiral perturbation theory and finite-volume partition functions (for a recent review see ). Within these results two different regimes have been investigated: (i) unscaled macroscopic correlations and (ii) microscopic correlations between eigenvalues rescaled by the mean spectral density. In the region (ii) one furthermore has to distinguish between scaling at the origin and in the bulk of the spectrum. In the macroscopic regime (i) the slope of the spectral density at the origin