Linear statistics on ensembles of random matrices occur frequently in many applications. We present a general method to compute probability distributions of linear statistics for large matrix size N. This is applied to the calculation of conductance and shot noise for ballistic scattering in chaotic cavities, in the limit of large number of open channels. The method is based on a mapping to a Coulomb gas problem in Laplace space, displaying phase transitions as the Laplace parameter is varied. As a consequence, the sough distribution displays a central Gaussian region flanked on both sides by non-Gaussian tails, and weak non-analytical points at the junction of the two regimes.
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