Euclidean random matrices: solved and open problems

TL;DR

This paper explores recent advances in understanding Euclidean random matrices, revealing phase transitions relevant to glassy materials and introducing a novel supersymmetric approach to analyze their properties.

Contribution

It presents new results on phase transitions in Euclidean random matrices and applies a supersymmetric method to study their spectral properties.

Findings

Identification of phonon-saddle phase transition in Euclidean matrices

Connection between phase transition and glass dynamics

Introduction of supersymmetric techniques for matrix analysis

Abstract

In this paper I will describe some results that have been recently obtained in the study of random Euclidean matrices, i.e. matrices that are functions of random points in Euclidean space. In the case of {\sl translation invariant} matrices one generically finds a phase transition between a {\sl phonon} phase and a {\sl saddle} phase. If we apply these considerations to the study of the Hessian of the Hamiltonian of the particles of a fluid, we find that this phonon-saddle transition corresponds to the dynamical phase transition in glasses, that has been studied in the framework of the mode coupling approximation. The Boson peak observed in glasses at low temperature is a remanent of this transition. We finally present some recent results obtained with a new approach where one deeply uses some hidden supersymmetric properties of the problem.