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Electronic transport through chaotic quantum dots exhibits universal, systemindependent, properties, consistent with random matrix theory. The quantumtransport can also be rooted, via the semiclassical approximation, in sums overthe classical scattering trajectories. Correlations between such trajectoriescan be organized diagrammatically and have been shown to yield universalanswers for some observables. Here, we develop the general combinatorialtreatment of the semiclassical diagrams, through a connection to factorizationsof permutations. We show agreement between the semiclassical and random matrixapproaches to the moments of the transmission eigenvalues. The result is validfor all moments to all orders of the expansion in inverse channel number forall three main symmetry classes (with and without time reversal symmetry andspin-orbit interaction) and extends to nonlinear statistics. This finallyexplains the applicability of random matrix theory to chaotic quantum transportin terms of the underlying dynamics as well as providing semiclassical accessto the probability density of the transmission eigenvalues.

Universality in chaotic quantum transport: The concordance between random matrix and semiclassical theories