Spectra of Euclidean Random Matrices

TL;DR

This paper investigates the spectral properties of Euclidean random matrices, introducing a field theory approach to develop a high-density expansion and approximate the spectrum, relevant for understanding fluid dynamics and glass transition phenomena.

Contribution

It presents a novel field theory framework for Euclidean random matrices, enabling systematic high-density expansions and spectrum approximations akin to the Coherent Potential Approximation.

Findings

Developed a field theory representation of Euclidean random matrices

Constructed a high-density expansion for the spectrum

Produced an approximation similar to the Coherent Potential Approximation

Abstract

We study the spectrum of a random matrix, whose elements depend on the Euclidean distance between points randomly distributed in space. This problem is widely studied in the context of the Instantaneous Normal Modes of fluids and is particularly relevant at the glass transition. We introduce a systematic study of this problem through its representation by a field theory. In this way we can easily construct a high density expansion, which can be resummed producing an approximation to the spectrum similar to the Coherent Potential Approximation for disordered systems.