Large-$N$ expansion for the time-delay matrix of ballistic chaotic cavities

TL;DR

This paper develops a recursive method to compute the $1/N$-expansion of moments of delay times in ballistic chaotic cavities using random matrix theory, applicable to systems with different symmetries.

Contribution

It introduces a recursion relation for the $1/N$-expansion coefficients of delay time moments, enhancing computational efficiency and understanding of their structure.

Findings

Derived recursion relations for $eta=1$ and $eta=2$ systems.

Confirmed integrality of expansion coefficients.

Discussed diagrammatic interpretation of coefficients.

Abstract

We consider the $1/N$-expansion of the moments of the proper delay times for a ballistic chaotic cavity supporting $N$ scattering channels. In the random matrix approach, these moments correspond to traces of negative powers of Wishart matrices. For systems with and without broken time reversal symmetry (Dyson indices $\beta=1$ and $\beta=2$) we obtain a recursion relation, which efficiently generates the coefficients of the $1/N$-expansion of the moments. The integrality of these coefficients and their possible diagrammatic interpretation is discussed.