We apply periodic orbit theory to a quantum billiard on a torus with avariable number N of small circular scatterers distributed randomly. Providedthese scatterers are much smaller than the wave length they may be regarded assources of diffraction. The relevant part of the spectral determinant is due todiffractive periodic orbits only. We formulate this diffractive zeta functionin terms of a N*N transfer matrix, which is transformed to real form. The zerosof this determinant can readily be computed. The determinant is shown toreproduce the full density of states for generic configurations if N>3. Westudy the statistics exhibited by these spectra. The numerical results suggestthat the spectra tend to GOE statistics as the number of scatterers increasesfor typical members of the ensemble. A peculiar situation arises forconfigurations with four scatterers and kR tuned to kR=y_{0,1}\approx 0.899,where the statistics appears to be perfectly Poissonian.
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