Clusters of near-degenerate levels dominate negative moments of spectral determinants

The negative moments of spectral determinants 0 as delta(-nu(k)). For a spectrum with equally distributed levels, the exponent nu(k) = k - 1. For random-matrix ensembles, with parameters beta = 1 (orthogonal), 2 (unitary), 4 (symplectic), we argue that the divergences for each k are determined by competitions between near-degenerate level clusters whose sizes depend on k, and we conjecture that nu(k) = int[(k - 1)/beta + 1]((k - 1 + 1/2beta) - 1/2beta int[(k - 1)/beta + 1]). For Poisson-distributed levels, unrestricted clustering leads to the delta-divergence of the moments increasing with the number N of levels in the interval considered, and nu(k) = N(k - 1).