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We study a random Schroedinger operator, the Laplacian with N independentlyuniformly distributed random delta potentials on flat tori T^d_L = R^d/LZ^d, d= 2, 3, where L > 0 is large. We determine a condition in terms of the size ofthe torus L, the density of the potentials \rho = N/L^d and the energy of theeigenfunction E such any such eigenfunctions will with nonzero probability beuniformly distributed on the entire torus. We remark that the equidistributionwe prove here is still consistent with a localized regime, where thelocalization length is much larger than the size of the torus. In fact ourresult implies a certain polynomial lower bound on the localization length, sothe localization length becomes infinitely large as E tends to infinity.

Uniformly distributed eigenfunctions on tori with random impurities