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

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

1993-04-07

arxiv.org/abs/hep-th/9304028v1

Physics

High Energy Physics

High Energy Physics - Theory

48 pages, IASSNS-HEP 93/14

Scientific paper

10.1007/BF02186812

We investigate the statistics of the number $N(R,S)$ of lattice points, $n\in \Z^2$, in a ``random'' annular domain $\Pi(R,w)=\,(R+w)A\,\setminus RA$, where $R,w >0$. Here $A$ is a fixed convex set with smooth boundary and $w$ is chosen so that the area of $\Pi (R,w)$ is $S$. The randomness comes from $R$ being taken as random ( with a smooth denisity ) in some interval $[c_1T,c_2T]$, $c_2>c_1>0$. We find that in the limit $T\to\infty $ the variance and distribution of $\De N=N(R;S)-S$ depends strongly on how $S$ grows with $T$. There is a saturation regime $S/T\to\infty$, as $T\to\infty$ in which the fluctuations in $\Delta N$ coming from the two boundaries of $\Pi $, are independent. Then there is a scaling regime, $S/T\to z$, $0

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