On random symmetric matrices with a constraint: the spectral density of random impedance networks

TL;DR

This paper derives the spectral density of large symmetric Gaussian random matrices with a row-sum constraint and relates it to the resonance density in random impedance networks, supported by analytical and numerical comparisons.

Contribution

It provides a new analytical expression for the eigenvalue density of constrained random matrices and links it to physical resonance phenomena in impedance networks.

Findings

Derived mean eigenvalue density for constrained matrices.

Established equivalence with resonance density in impedance networks.

Validated results with numerical simulations.

Abstract

We derive the mean eigenvalue density for symmetric Gaussian random N x N matrices in the limit of large N, with a constraint implying that the row sum of matrix elements should vanish. The result is shown to be equivalent to a result found recently for the average density of resonances in random impedance networks [Y.V. Fyodorov, J. Phys. A: Math. Gen. 32, 7429 (1999)]. In the case of banded matrices, the analytical results are compared with those extracted from the numerical solution of Kirchhoff equations for quasi one-dimensional random impedance networks.