Loop Representations of the Quark Determinant in Lattice QCD

The modelling of the ultraviolet contributions to the quark determinant in lattice QCD in terms of a small number of Wilson loops is examined. Complete Dirac spectra are obtained for sizeable ensembles of SU(3) gauge fields at $\beta$=5.7 on 6$^4$, 8$^4$, and 10$^4$ lattices allowing for the first time a detailed study of the volume dependence of the effective loop action generating the quark determinant. The connection to the hopping parameter expansion is examined in the heavy quark limit. We compare the efficiency and accuracy of various methods- specifically, Lanczos versus stochastic approaches- for extracting the quark determinant on an ensemble of configurations.