Statistics of time delay and scattering correlation functions in chaotic systems I. Random Matrix Theory

We consider the statistics of time delay in a chaotic cavity having $M$ open channels, in the absence of time-reversal invariance. In the random matrix theory approach, we compute the average value of polynomial functions of the time delay matrix $Q=-i\hbar S^\dag dS/dE$, where $S$ is the scattering matrix. Our results do not assume $M$ to be large. In a companion paper, we develop a semiclassical approximation to $S$-matrix correlation functions, from which the statistics of $Q$ can also be derived. Together, these papers contribute to establishing the conjectured equivalence between the random matrix and the semiclassical approaches.
Comment: 9 pages, no figures. A previous paper, arXiv:1408.1669, has been split in two parts upon publication, each part containing different results, in order to make the presentation more consistent. This is the first part