Euclidean Random Matrices Research Articles

Properties of stable ensembles of Euclidean random matrices.

We study the spectrum of a system of coupled disordered harmonic oscillators in the thermodynamic limit. This Euclidean random matrix ensemble has been suggested as a model for the low temperature vibrational properties of glass. Exact numerical diagonalization is performed in three and two spatial dimensions, which is accompanied by a detailed finite size analysis. It reveals a low-frequency regime of sound waves that are damped by Rayleigh scattering. At large frequencies localized modes exist. In between, the central peak in the vibrational density of states is well described by Wigner's semicircle law for not too large disorder, as is expected for simple random matrix systems. We compare our results with predictions from two recent self-consistent field theories.

Spectrum of large euclidean random matrices generated from l p ellipsoids

In this paper, we study the probability distribution of eigenvalues of the Euclidean random matrix whose entry is of the form for some real function f. Random points 's are independently distributed in the ellipsoid or on its surface in including the unit sphere , the simplex, the ordinary ellipsoid and the hyper-cube. Here is allowed to be any real number which includes the two most interesting cases and . The limits of the empirical distribution of its eigenvalues are derived in two high dimensional settings: and as both n and N go to infinity. By taking to be suitable functions, we also give the explicit limiting spectral distributions for some distance matrices whose entries are based on the Euclidean distance and the geodesic distance.

Spectrum of multispace Euclidean random matrices.

We consider the additive superimposition of an extensive number of independent Euclidean Random Matrices in the high-density regime. The resolvent is computed with techniques from free probability theory, as well as with the replica method of statistical physics of disordered systems. Results for the spectrum and eigenmodes are shown for a few applications relevant to computational neuroscience, and are corroborated by numerical simulations.

Renormalization to localization without a small parameter

We study the wave function localization properties in a d-dimensional model of randomly spaced particles with isotropic hopping potential depending solely on Euclidean interparticle distances. Due to generality of this model usually called Euclidean random matrix model, it arises naturally in various physical contexts such as studies of vibrational modes, artificial atomic systems, liquids and glasses, ultracold gases and photon localization phenomena. We generalize the known Burin-Levitov renormalization group approach, formulate universal conditions sufficient for localization in such models and inspect a striking equivalence of the wave function spatial decay between Euclidean random matrices and translation-invariant long-range lattice models with a diagonal disorder.

Random Coulomb antiferromagnets: From diluted spin liquids to Euclidean random matrices

We study a disordered classical Heisenberg magnet with uniformly antiferromagnetic interactions which are frustrated on account of their long-range Coulomb form, {\em i.e.} $J(r)\sim -A\ln r$ in $d=2$ and $J(r)\sim A/r$ in $d=3$. This arises naturally as the $T\rightarrow 0$ limit of the emergent interactions between vacancy-induced degrees of freedom in a class of diluted Coulomb spin liquids (including the classical Heisenberg antiferromagnets on checkerboard, SCGO and pyrochlore lattices) and presents a novel variant of a disordered long-range spin Hamiltonian. Using detailed analytical and numerical studies we establish that this model exhibits a very broad paramagnetic regime that extends to very large values of $A$ in both $d=2$ and $d=3$. In $d=2$, using the lattice-Green function based finite-size regularization of the Coulomb potential (which corresponds naturally to the underlying low-temperature limit of the emergent interactions between orphan-spins), we only find evidence that freezing into a glassy state occurs in the limit of strong coupling, $A=\infty$, while no such transition seems to exist at all in $d=3$. We also demonstrate the presence and importance of screening for such a magnet. We analyse the spectrum of the Euclidean random matrices describing a Gaussian version of this problem, and identify a corresponding quantum mechanical scattering problem.

Metapopulation persistence in random fragmented landscapes.

Habitat destruction and land use change are making the world in which natural populations live increasingly fragmented, often leading to local extinctions. Although local populations might undergo extinction, a metapopulation may still be viable as long as patches of suitable habitat are connected by dispersal, so that empty patches can be recolonized. Thus far, metapopulations models have either taken a mean-field approach, or have modeled empirically-based, realistic landscapes. Here we show that an intermediate level of complexity between these two extremes is to consider random landscapes, in which the patches of suitable habitat are randomly arranged in an area (or volume). Using methods borrowed from the mathematics of Random Geometric Graphs and Euclidean Random Matrices, we derive a simple, analytic criterion for the persistence of the metapopulation in random fragmented landscapes. Our results show how the density of patches, the variability in their value, the shape of the dispersal kernel, and the dimensionality of the landscape all contribute to determining the fate of the metapopulation. Using this framework, we derive sufficient conditions for the population to be spatially localized, such that spatially confined clusters of patches act as a source of dispersal for the whole landscape. Finally, we show that a regular arrangement of the patches is always detrimental for persistence, compared to the random arrangement of the patches. Given the strong parallel between metapopulation models and contact processes, our results are also applicable to models of disease spread on spatial networks.

Eigenspectrum bounds for semirandom matrices with modular and spatial structure for neural networks.

The eigenvalue spectrum of the matrix of directed weights defining a neural network model is informative of several stability and dynamical properties of network activity. Existing results for eigenspectra of sparse asymmetric random matrices neglect spatial or other constraints in determining entries in these matrices, and so are of partial applicability to cortical-like architectures. Here we examine a parameterized class of networks that are defined by sparse connectivity, with connection weighting modulated by physical proximity (i.e., asymmetric Euclidean random matrices), modular network partitioning, and functional specificity within the excitatory population. We present a set of analytical constraints that apply to the eigenvalue spectra of associated weight matrices, highlighting the relationship between connectivity rules and classes of network dynamics.

Distribution of eigenvalues of large Euclidean matrices generated from ellipsoid

Abstract In this paper, we study large Euclidean random matrices M n = ( f n ( ‖ x i − x j ‖ 2 ) ) n × n where x i ’s are i.i.d. points l p -norm uniformly distributed over the N dimensional l p ellipsoid or its surface, f n is a real function on [ 0 , ∞ ) and ‖ ⋅ ‖ is the Euclidean distance. Under the assumption that both N and n go to infinity proportionally, we obtain the limiting empirical distribution of the eigenvalues of M n . The limit is closely related to the semi-axis lengths of the l p ellipsoid or its surface.

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

In this paper we will first study the spectrum of certain large Euclidean random matrices. The entries of these matrices are functions of n random points lp-norm uniformly sampled in N dimensional lp ball or lp sphere. Under the setting n/N → 0, as N and n tend to ∞, it is shown that the empirical distributions of eigenvalues of normalized Euclidean random matrices generated from the above two manifolds both converge weakly to semicircle law. We also investigate certain ‘sample-covariance’ type matrices whose entries are determined by n random positions from an N dimensional lp ellipsoid or its surface. When N → ∞ and n → ∞ with N/n → 0, we prove that their empirical spectral distribution converge to the same fixed distribution.

On the high-density expansion for Euclidean random matrices

Diagrammatic techniques to compute perturbatively the spectral properties of Euclideanrandom matrices (ERM) in the high-density regime are introduced and discussed in detail.Such techniques are developed in two alternative and very different formulations of themathematical problem and are shown to give identical results up to second order in theperturbative expansion. One method, based on writing the so-called resolventfunction as a Taylor series, allows us to group the diagrams into a small number oftopological classes, providing a simple way to determine the infrared (small momenta)behaviour of the theory up to third order, which is of interest for the comparisonwith experiments. The other method, which reformulates the problem as a fieldtheory, can instead be used to study the infrared behaviour at any perturbativeorder.

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

We study probability distributions of eigenvalues of Hermitian and non-Hermitian Euclidean random matrices that are typically encountered in the problems of wave propagation in random media.

Localization-delocalization transition in Hessian matrices of topologically disordered systems

Using the level-spacing (LS) statistics, we have investigated the localization-delocalization transitions (LDTs) in Hessian matrices of a simple fluid with short-ranged interactions. The model fluid is a prototype of topologically disordered systems and its Hessian matrices are recognized as an ensemble of Euclidean random matrices with elements subject to several kinds of constraints. Two LDTs in the Hessian matrices are found, with one in the positive-eigenvalue branch and the other in the negative-eigenvalue one. The locations and the critical exponents of the two LDTs are estimated by the finite-size scaling for the second moments of the nearest-neighbor LS distributions. Within numerical errors, the two estimated critical exponents are almost coincident with each other and close to that of the Anderson model (AM) in three dimensions. The nearest-neighbor LS distribution at each LDT is examined to be in a good agreement with that of the AM at the critical disorder. We conclude that the LDTs in the Hessian matrices of topologically disordered systems exhibit the critical behaviors of orthogonal universality class.

Eigenvalues of Euclidean random matrices

AbstractWe study the spectral measure of large Euclidean random matrices. The entries of these matrices are determined by the relative position of n random points in a compact set Ωn of ℝd. Under various assumptions, we establish the almost sure convergence of the limiting spectral measure as the number of points goes to infinity. The moments of the limiting distribution are computed, and we prove that the limit of this limiting distribution as the density of points goes to infinity has a nice expression. We apply our results to the adjacency matrix of the geometric graph. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008

Anderson localization in Euclidean random matrices

We study the spectra and localization properties of Euclidean random matrices defined on a random graph. We introduce a powerful method to find the density of states and the localization threshold in topologically disordered (off-lattice) systems. We solve numerically an equation (exact on the random graph) for the probability distribution function of the diagonal elements of the resolvent matrix, with a population dynamics algorithm (PDA). We show that the localization threshold can be estimated by studying the stability of a population of real resolvents under the PDA. An application is given in the context of the instantaneous normal modes of a liquid.

Euclidean random matrices, the glass transition and the boson peak

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.

Vibrational spectra in glasses

Abstract The findings of X-ray and neutron scattering experiments on amorphous systems are interpreted within the framework of the theory of Euclidean random matrices. This allows us to take into account the topological nature of the disorder, a key ingredient which strongly affects the vibrational spectra of those systems. We present a resummation scheme for a perturbative expansion in the inverse particle density, allowing an accurate analytical computation of the dynamic structure factor within the range of densities encountered in real systems.

The dynamical structure factor in topologically disordered systems

A computation of the dynamical structure factor of topologically disordered systems, where the disorder can be described in terms of Euclidean random matrices, is presented. Among others, structural glasses and supercooled liquids belong to that class of systems. The computation describes their relevant spectral features in the region of the high frequency sound. The analytical results are tested with numerical simulations and are found to be in very good agreement with them. Our results may explain the findings of inelastic x-ray scattering experiments in various glassy systems.

Vibrational spectra in glasses

The findings of X-ray and neutron scattering experiments on amorphous systems are interpreted within the framework of the theory of Euclidean random matrices. This allows to take into account the topological nature of the disorder, a key ingredient which strongly affects the vibrational spectra of those systems. We present a resummation scheme for a perturbative expansion in the inverse particle density, allowing an accurate analytical computation of the dynamical structure factor within the range of densities encountered in real systems.

Hopping between random locations: spectrum andinstanton

Euclidean random matrices appear in a broad class of physicalproblems involving disorder. The problem of determining theirspectra can be mapped, using the replica method, into the study of ascalar field theory with an interaction of the type eψ2.We apply the instanton method to study their spectral tails.