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

TL;DR

This paper analyzes the statistical properties of time delay in chaotic systems using Random Matrix Theory, providing exact results without large channel assumptions, and aims to connect with semiclassical methods.

Contribution

It computes the average polynomial functions of the time delay matrix in chaotic systems without assuming a large number of channels, bridging RMT and semiclassical approaches.

Findings

Derived exact statistical properties of time delay matrices

Established connections between RMT and semiclassical methods

Provided results valid for any number of open channels

Abstract

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.