Statistical properties of the spectrum of the QCD Dirac operator at low energy

We analyze the statistical properties of the spectrum of the QCD Dirac oper ator at low energy in a finite box of volume L-4 by means of partially quen ched Chiral Perturbation Theory, a low-energy effective field theory based on the symmetries of QCD. We derive the two-point spectral correlation func tion from the discontinuity of the chiral susceptibility. For eigenvalues m uch smaller than m(c) = F-2/SigmaL(2), where F is the pion decay constant a nd C is the absolute value of the quark condensate, our result For the two- point correlation function coincides with the result previously obtained fr om chiral Random Matrix Theory (chRMT). The departure from the chRMT result above that scale is due to the contribution of the nonzero momentum modes. In terms of the variance oft he number of eigenvalues in an interval conta ining n eigenvalues on average, it amounts to a crossover from a log n-beha vior to a n(2) log n-behavior. (C) 2001 Elsevier Science B.V. All rights re served.