Universal Frequency Correlations and Recurrence Statistics of Complex Impedance Matrices

TL;DR

This paper investigates the universal statistical properties of impedance matrices in complex electromagnetic scattering systems, providing numerical and experimental evidence for frequency correlations and recurrence behaviors across various physical platforms.

Contribution

It introduces universal scaling laws for impedance matrix correlations and recurrence intervals, validated through experiments on microwave graphs, billiards, and cavities.

Findings

Impedance matrix elements exhibit universal frequency correlation functions.

Repetition intervals of impedance values follow predictable statistical patterns.

Experimental results align well with theoretical predictions.

Abstract

Linear electromagnetic wave scattering systems can be characterized by an impedance matrix that relates the voltages and currents at the ports of the system. When the system size becomes greater than the wavelength of the fields involved, the impedance matrix becomes a complicated function of the details of the system, in which case a statistical model, such as the Random Coupling Model (RCM) becomes useful. The statistics of the elements of the RCM impedance matrix depend on the excitation frequency, the spectral density of the modes of the enclosed system volume, the average loss factor (Q^{-1}) of the system, and the properties of the coupling ports as given by their radiation impedances. In this paper, properties of the elements of impedance matrices are explored numerically and experimentally. These include the two point frequency correlation functions for the complex impedance of elements and the expected difference in frequencies between which impedance values are approximately repeated. Universal scaling arguments are then given for these quantities, hence these results are generic for all sufficiently complicated scattering systems, including acoustic and optical systems. The experimental data presented in this paper come from microwave graphs, billiards, and three-dimensional cavities with embedded tunable perturbers such as metasurfaces. The data is found to be in generally good agreement with the predictions for the two point frequency correlations and the frequency interval for successive repetitions of impedance matrix elements values.