Statistics of resonances and delay times: A criterion for Metal-Insulator transitions

We study the distributions of the normalized resonance widths ${\cal P} ({\tilde \Gamma})$anddelaytimes${\cal P} ({\tilde \tau})$for$3$D disordered tight-binding systems at the metal-insulator transition (MIT) by attaching leads to the boundary sites. Both distributions are scale invariant, independent of the microscopic details of the random potential, and the number of channels. Theoretical considerations suggest the existence of a scaling theory for $\mathcal{P}(\tilde{\mathrm{\Gamma }})$ in finite samples, and numerical calculations confirm this hypothesis. Based on this, we give a new criterion for the determination and analysis of the MIT.