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Joint statistics of quantum transport in chaotic cavities

We study the joint statistics of conductance $G$ and shot noise $P$ in chaotic cavities supporting a large number $N$ of open electronic channels in the two attached leads. We determine the full phase diagram in the $(G,P)$ plane, employing a Coulomb gas technique on the joint density of transmission eigenvalues, as dictated by Random Matrix Theory. We find that in the region of typical fluctuations, conductance and shot noise are uncorrelated and jointly Gaussian, and away from it they fluctuate according to a different joint rate function in each phase of the $(G,P)$ plane. Different functional forms of the rate function in different regions emerge as a direct consequence of third order phase transitions in the associated Coulomb gas problem.