Spectral statistic for decaying random potentials

We consider Anderson model $H^{\omega }=-\mathrm{\Delta }+V^{\omega }$ on ${\mathrm{\ell }}^2({\mathbb{Z}}^d)$ with decaying random potential. We study the point process ${\xi }_{L,\lambda }^{\omega }$ associated with eigenvalues of $H_{{\mathrm{\Lambda }}_L}^{\omega }$, the retriction of $H^{\omega }$ to the finite cube ${\mathrm{\Lambda }}_L$. Our result is that the weak limit points of $\{{\xi }_{L,\lambda }^{\omega }\}$ are poisson point processes as $L\to \mathrm{\infty }$.