We study statistical properties of the ensemble of large NϫN random matrices whose entries H i j decrease in a power-law fashion H i j ϳ͉iϪ j͉ Ϫ␣ . Mapping the problem onto a nonlinear model with nonlocal interaction, we find a transition from localized to extended states at ␣ϭ1. At this critical value of ␣ the system exhibits multifractality and spectral statistics intermediate between the Wigner-Dyson and Poisson statistics. These features are reminiscent of those typical of the mobility edge of disordered conductors. We find a continuous set of critical theories at ␣ϭ1, parametrized by the value of the coupling constant of the model. At ␣Ͼ1 all states are expected to be localized with integrable power-law tails. At the same time, for 1Ͻ␣Ͻ3/2 the wave packet spreading at a short time scale is superdiffusive: ͉͗r͉͘ϳt 1/(2␣Ϫ1) , which leads to a modification of the Altshuler-Shklovskii behavior of the spectral correlation function. At 1/2Ͻ␣Ͻ1 the statistical properties of eigenstates are similar to those in a metallic sample in dϭ(␣Ϫ1/2) Ϫ1 dimensions. Finally, the region ␣Ͻ1/2 is equivalent to the corresponding Gaussian ensemble of random matrices (␣ϭ0). The theoretical predictions are compared with results of numerical simulations.

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