Energy-Level Statistics of Model Quantum Systems: Universality and Scaling in a Lattice-Point Problem

We investigate the statistics of the number N(R, S) of lattice points, n E Z-', in an annular domain /7(R, w) = (R + w)A\RA, where R, w > 0. Here A is a fixed convex set with smooth boundary and w is chosen so that the area of/7(R, w) is S. The statistics comes from R being taken as random (with a smooth density) in some interval [c~ T, c, T], c2 > ct > 0. We find that in the limit T--* oo the variance and distribution of A N = N(R; S ) S depend strongly on how S grows with T. There is a saturation regime S / T , oo, as T-+ co, in which the fluctuations in dN coming from the two boundaries o f /7 are independent. Then there is a scaling regime, S/T---, z, 0 < z < oo, in which the distribution depends on z in an almost periodic way going to a Gaussian as z ~ 0. The variance in this limit approaches z for "generic" A, but can be larger for "degenerate" cases. The former behavior is what one would expect from the Poisson limit of a distribution for annuli of finite area.