Level and Eigenfunction Statistics in Billiards with Surface Scattering
  Statistical properties of billiards with diffusive boundary scattering are investigated by means of the supersymmetric sigma-model in a formulation appropriate for chaotic ballistic systems. We study level statistics, parametric level statistics, and properties of electron wavefunctions. In the universal regime, our results reproduce conclusions of the random matrix theory, while beyond this regime we obtain a variety of system-specific results determined by the classical dynamics in the billiard. Most notably, we find that level correlations do not vanish at arbitrary separation between energy levels, or if measured at arbitrarily large difference of magnetic fields. Saturation of the level number variance indicates strong rigidity of the spectrum. To study spatial correlations of wavefunction amplitudes, we reanalyze and refine derivation of the ballistic version of the sigma-model. This allows us to obtain a proper matching of universal short-scale correlations with system-specific ones. 
Ya. M. Blanter, A. D. Mirlin, and B. A. Muzykantskii