Statistics of quantum transport in chaotic cavities with broken time-reversal symmetry

The statistical properties of quantum transport through a chaotic cavity are encoded in the traces ${\mathcal{T}}_{n}=\text{Tr}[{(t{t}^{\ifmmode\dagger\else\textdagger\fi{}})}^{n}]$, where $t$ is the transmission matrix. Within the random matrix theory approach, these traces are random variables whose probability distribution depends on the symmetries of the system. For the case of broken time-reversal symmetry, we use generalizations of Selberg's integral and the theory of symmetric polynomials to present explicit closed expressions for the average value, and for the variance of ${\mathcal{T}}_{n}$ for all $n$. In particular, this provides the charge cumulants $⟨⟨{Q}_{n}⟩⟩$ of all orders. We also compute the moments $⟨{g}^{n}⟩$ of the conductance $g={\mathcal{T}}_{1}$. All the results obtained are exact, i.e., they are valid for arbitrary numbers of open channels.