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

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 $Q=-i\hslash S^{†}dS/dE$, 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.

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

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• Correlators for the Wigner-Smith time-delay matrix of chaotic cavities
• Large-$N$ expansion for the time-delay matrix of ballistic chaotic cavities
• Wigner-Smith time-delay matrix in chaotic cavities with non-ideal contacts
• Statistics of time delay and scattering correlation functions in chaotic systems II. Semiclassical Approximation
• Chaotic Scattering with Localized Losses: S-Matrix Zeros and Reflection Time Difference for Systems with Broken Time Reversal Invariance
• Truncated linear statistics associated with the eigenvalues of random matrices II. Partial sums over proper time delays for chaotic quantum dots
• Eigenfunction non-orthogonality factors and the shape of CPA-like dips in a single-channel reflection from lossy chaotic cavities
• A Physical Interpretation of Imaginary Time Delay
• Semiclassical approach to $S$ matrix energy correlations and time delay in chaotic systems
• Superuniversal Statistics of Complex Time-Delays in Non-Hermitian Scattering Systems
• Delay times in chaotic quantum systems
• Schur expansion of random-matrix reproducing kernels