Eur. Phys. J. B 19, 121-132 (2001)
https://doi.org/10.1007/s100510170357

Short-range plasma model for intermediate spectral statistics

¹ Laboratoire de Physique Théorique et Modèles Statistiques (Unité de Recherche de l'Université Paris XI et du CNRS (UMR 8626)) , Université Paris-Sud, 91405 Orsay Cedex, France
² Phys. Dep., UCSD, La Jolla, CA 92093-0319, USA

Corresponding author: ^a bogomol@ipno.in2p3.fr

Received: 13 September 2000
Published online: 15 January 2001

Abstract

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 for large L and the nearest-neighbor distribution decreases exponentially when , with , where β is the inverse temperature of the gas (, 2 and 4 for the orthogonal, unitary and symplectic symmetry class respectively). In the simplest case of , the model leads to the so-called Semi-Poisson statistics characterized by particular simple correlation functions e.g. . Furthermore we investigate the spectral statistics of several pseudointegrable quantum billiards numerically and compare them to the Semi-Poisson statistics.