Spectral Properties of Random Reactance Networks and Random Matrix Pencils

TL;DR

This paper investigates the spectral properties of eigenvalues of random matrix pencils arising from large random LC networks, revealing that their fluctuations follow Wigner-Dyson statistics and providing detailed spectral density calculations.

Contribution

It introduces a comprehensive analysis of eigenvalue distributions of random matrix pencils in LC networks, applying supersymmetry techniques to derive spectral statistics.

Findings

Eigenvalue density and correlation functions are explicitly calculated.

Spectral fluctuations match Wigner-Dyson universality.

Results apply to both finite and infinite-range network models.

Abstract

Our goal is to study statistical properies of "dielectric resonances" which are poles of conductance of a large random $LC$ network. Such poles are a particular example of eigenvalues $\lambda_n$ of matrix pencils ${\bf H}-\lambda {\bf W}$, with ${\bf W}$ being positive definite matrix and ${\bf H}$ a random real symmetric one. We first consider spectra of matrix pencils with independent, identically distributed entries of ${\bf H}$. Then we concentrate on an infinite-range ("full-connectivity") version of random $LC$ network. In all cases we calculate the mean eigenvalue density and the two-point correlation function in the framework of Efetov's supersymmetry approach. Fluctuations in spectra turn out to be the same as those provided by Wigner-Dyson theory of usual random matrices.