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

TL;DR

This paper studies the statistical behavior of lattice points in random annular regions, revealing different fluctuation regimes and their dependence on the size of the region relative to the scaling parameter.

Contribution

It introduces a detailed analysis of lattice point fluctuations in random annuli, identifying saturation and scaling regimes with explicit variance behavior.

Findings

Variance depends on the growth rate of S with T.

In the saturation regime, fluctuations are independent.

In the scaling regime, the distribution approaches a Gaussian as z approaches zero.

Abstract

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<z<\infty $ in which the distribution depends on $z$ in an almost periodic way going to a Gaussian as $z\to\ 0$. 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.