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Transport moments beyond the leading order

For chaotic cavities with scattering leads attached, transport properties can be approximated in terms of the classical trajectories which enter and exit the system. With a semiclassical treatment involving fine correlations between such trajectories we develop a diagrammatic technique to calculate the moments of various transport quantities. Namely, we find the moments of the transmission and reflection eigenvalues for systems with and without time reversal symmetry. We also derive related quantities involving an energy dependence: the moments of the Wigner delay times and the density of states of chaotic Andreev billiards, where we find that the gap in the density persists when subleading corrections are included. Finally, we show how to adapt our techniques to non-linear statistics by calculating the correlation between transport moments. In each setting, the answer for the $n$-th moment is obtained for arbitrary $n$ (in the form of a moment generating function) and for up to the three leading orders in terms of the inverse channel number. Our results suggest patterns which should hold for further corrections and by matching with the low order moments available from random matrix theory we derive likely higher order generating functions.