Short-range plasma model for intermediate spectral statistics

We propose a plasma model for spectral statistics displaying level repulsion without long-range spectral rigidity, i.e. statistics intermediate between random matrix and Poisson statistics similar to the ones found numerically at the critical point of the Anderson metal-insulator transition in disordered systems and in certain dynamical systems. The model emerges from Dysons one-dimensional gas corresponding to the eigenvalue distribution of the classical random matrix ensembles by restricting the logarithmic pair interaction to a finite number k of nearest neighbors. We calculate analytically the spacing distributions and the two-level statistics. In particular we show that the number variance has the asymptotic form Sigma (2)(L) similar to chiL for large L and the nearest-neighbor distribution decreases exponentially when s --> infinity, P(s) similar to exp(-Lambdas) with Lambda = 1/chi = k beta + 1, where beta is the inverse temperature of the gas (beta = 1, 2 and 4 for the orthogonal, unitary and symplectic symmetry class respectively). In the simplest case of k = beta = 1, the model leads to the so-called Semi-Poisson statistics characterized by particular simple correlation functions e.g. P(s) = 4s exp(-2s). Furthermore we investigate the spectral statistics of several pseudointegrable quantum billiards numerically and compare them to the Semi-Poisson statistics.