Euclidean Randommatrices:solvedand Open Problems

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 translation invariant matrices one generically finds a phase transition between a phonon phase and a 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.

Eigenvalue distributions of large Euclidean random matrices for waves in random media

“…Despite the differences between the matrices Ĝ and X, a comparison of their eigenvalue distributions appears to be quite useful. Numerical calculations show that, roughly speaking, the eigenvalues of Ĝ are concentrated within a circle on the complex plane (see figures [6][7][8]. The same circular shape of the domain of existence of eigenvalues is the characteristic of the matrix X in the limit of β → 0.…”

“…We thus conclude that in the limit of β 1, the domains of existence of eigenvalues of the matrices Ĝ and X are very similar. In addition to the eigenvalues inside the circle, Ĝ has eigenvalues that follow the spirals corresponding to the eigenvalues G 12 and −G 12 of a 2 × 2 matrix Ĝ. Interestingly, the spirals are quite robust and survive at all densities (see figures [6][7][8].…”

“…. , N) are randomly chosen points in the Euclidean space [5,6]. Euclidean random matrices appear in various physical contexts and were previously considered to interpret the 'boson peak' in supercooled liquids [7] or to study slow relaxation in glasses and scalar phonon localization [8], to cite a few recent examples.…”

Cavity approach to the spectral density of sparse symmetric random matrices

“…In writing the expression (3), we have been careless with the Gaussian integrals, so that as they stand they are not generally convergent; we simply follow the prescription as in [14,15], and do not worry unnecessarily about imaginary factors, so that we can introduce a GibbsBoltzmann probability distribution of x, viz.…”

Distributions for the eigenvalues of large random matrices generated from four manifolds

“…. , n. Euclidean random matrices play an important role in the description of many physical models, including: the electronic levels in amorphous systems, very diluted impurities, and the spectrum of vibrations in glasses ( [15,21,23,25]). In this paper, we first focus on a special class of Hermitian Euclidean random matrices, taking the form…”