On the high-density expansion for Euclidean Random Matrices

TL;DR

This paper introduces diagrammatic techniques for perturbative analysis of Euclidean Random Matrices at high density, providing two formulations that yield consistent results and facilitate understanding of their spectral properties.

Contribution

It develops two distinct diagrammatic methods for analyzing Euclidean Random Matrices perturbatively, enabling detailed study of their spectral properties and infrared behavior.

Findings

Methods agree up to second order in perturbation

Topological classification simplifies infrared analysis

Field theory approach allows all-order infrared behavior study

Abstract

Diagrammatic techniques to compute perturbatively the spectral properties of Euclidean Random Matrices in the high-density regime are introduced and discussed in detail. Such techniques are developed in two alternative and very different formulations of the mathematical problem and are shown to give identical results up to second order in the perturbative expansion. One method, based on writing the so-called resolvent function as a Taylor series, allows to group the diagrams in a small number of topological classes, providing a simple way to determine the infrared (small momenta) behavior of the theory up to third order, which is of interest for the comparison with experiments. The other method, which reformulates the problem as a field theory, can instead be used to study the infrared behaviour at any perturbative order.