Scattering phase of quantum dots: Emergence of universal behavior*

We investigate scattering through chaotic ballistic quantum dots in the Coulomb-blockade regime. Focusing on the scattering phase, we show that long universal sequences emerge in the short wavelength limit of many electrons on the dot, where phase lapses of $\pi$ systematically occur in between two consecutive resonances. We further argue that such universal sequences become shorter and shorter as the wavelength becomes larger/the number of electrons on the dot is reduced. Our results are corroborated by numerics and are in qualitative and quantitative agreement with experimental results. We finally present numerical data on models of interacting electrons to show that strong correlations do not alter our conclusions.