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

Skipetrov, S. E.; Goetschy, Arthur

doi:10.1088/1751-8113/44/6/065102

Cited by 66 publications

(118 citation statements)

References 45 publications

(227 reference statements)

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“…Random matrices that incorporate this constraint are known as Euclidean random matrices [28], and have been used to model physical phenomena such as diffusion [29] and wave propagation [30]. Neurons in the neocortex reside in physical space, with connection probabilities modulated smoothly across the cortical surface [31][32][33].…”

Section: B Spatial and Functional Connectivity Constraintsmentioning

confidence: 99%

“…Previous results have derived Eigenvalue spectra for Hermitian and non-Hermitian Euclidean random matrices by forming analytical decompositions of the spatial operator resolvent [30,45], but have not analyzed matrices with block-signed structure similar to those we discuss here. Our results for networks with smooth spatial and functional connectivity indicate that in Euclidean random matrices including a signed bipartition of positive and negative elements the bound of the eigenspectrum becomes increasingly sensitive to the local balance between positive and negative interactions as the spatial range of interactions decreases.…”

Section: Implications For Dynamics and Computation In Neural Networkmentioning

confidence: 99%

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

Muir

Mrsic-Flogel

2015

Phys. Rev. E

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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.

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Section: B Spatial and Functional Connectivity Constraintsmentioning

confidence: 99%

Section: Implications For Dynamics and Computation In Neural Networkmentioning

confidence: 99%

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

Muir

Mrsic-Flogel

2015

Phys. Rev. E

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“…While tempting, it is not possible to interpolate between the 1-stable and universal Gaussian RMT by introducing a high-energy cutoff to the density (17), since the statistics of (infinitely) large matrices with a truncated Lévy distribution always lie in the basin of attraction of the GOE and therefore make exactly the same predictions as universal Gaussian RMT. However, it might be possible to treat the intermediate case within the general theory of Euclidean random matrices (ERMT) [63,65,[93][94][95][96][97][98] that addresses all those very large N × N matrices M the elements M ij of which depend on pairs R i , R j of N randomly chosen coordinates-precisely as for the Rydberg Hamiltonian, Eq. (4).…”

Section: Stable Random Matricesmentioning

confidence: 99%

Spectral backbone of excitation transport in ultracold Rydberg gases

2014

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The spectral structure underlying excitonic energy transfer in ultra-cold Rydberg gases is studied numerically, in the framework of random matrix theory, and via self-consistent diagrammatic techniques. Rydberg gases are made up of randomly distributed, highly polarizable atoms that interact via strong dipolar forces. Dynamics in such a system is fundamentally different from cases in which the interactions are of short range, and is ultimately determined by the spectral and eigenvector structure. In the energy levels' spacing statistics, we find evidence for a critical energy that separates delocalized eigenstates from states that are localized at pairs or clusters of atoms separated by less than the typical nearest-neighbor distance. We argue that the dipole blockade effect in Rydberg gases can be leveraged to manipulate this transition across a wide range: As the blockade radius increases, the relative weight of localized states is reduced. At the same time, the spectral statistics-in particular, the density of states and the nearest neighbor level spacing statistics-exhibits a transition from approximately a 1-stable Lévy to a Gaussian orthogonal ensemble. Deviations from random matrix statistics are shown to stem from correlations between inter-atomic interaction strengths that lead to an asymmetry of the spectral density and profoundly affect localization properties. We discuss approximations to the self-consistent Matsubara-Toyozawa locator expansion that incorporate these effects.

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“…1(b)]. In particular, the m = ±1 groups of eigenvalues develop "holes" that were previously associated with Anderson localization in the framework of the scalar model of wave scattering [32,33].…”

mentioning

confidence: 91%

Magnetic-Field-Driven Localization of Light in a Cold-Atom Gas

2015

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We discover a transition from extended to localized quasi-modes for light in a gas of immobile two-level atoms in a magnetic field. The transition takes place either upon increasing the number density of atoms in a strong field or upon increasing the field at a high enough density. It has many characteristic features of a disorder-driven (Anderson) transition but is strongly influenced by near-field interactions between atoms and the anisotropy of the atomic medium induced by the magnetic field.The transition from extended to localized eigenstates upon increasing disorder in a quantum or wave system is called after Philip Anderson who was the first to predict it for electrons in disordered solids [1]. More recently, this transition was studied for various types of quantum particles (cold atoms [2], Bose-Einstein condensates [3]) as well as for classical waves (light [4-6], ultrasound [7,8]). In the most common case of time-reversal symmetric systems invariant under spin rotation Anderson transition takes place for a three-dimensional (3D) disorder only, eigenstates of low-dimensional systems being always localized [9,10]. Anderson localization of light may find applications in the design of future quantuminformation devices [11], miniature lasers [12] and solar cells [13]. However, no undisputable experimental observation of optical Anderson transition in 3D exists to date since alternative explanations were proposed for all published reports of it [14][15][16]. Moreover, we have recently shown that the simplest theoretical model in which light is scattered by point scatterers (atoms) does not predict Anderson localization of light at all [17].In the present Letter we show that an external magnetic field may induce a transition between extended and localized states for light in a gas of cold atoms. Magnetic field is an important and unique means of controlling wave propagation in disordered media. On the one hand, it breaks down the time-reversal invariance leading to a suppression of weak localization in electronic [18] and optical [19] systems and to metal-insulator transitions in topological insulators [20]. On the other hand, by profoundly modifying the scattering properties of individual scatterers the magnetic field can produce an enhancement of the coherent backscattering peak for light scattered by atoms with a degenerate ground state [21,22]. Our work adds a new element in the mosaic of magnetic-field-induced phenomena in disordered systems by demonstrating that the removal of degeneracy of the excited atomic state due to the Zeeman effect and the resulting reduction of the strength of resonant dipole-dipole * Sergey. We consider an ensemble of N 1 identical two-level atoms at random position {r i } inside a spherical volume V of radius R. The resonant frequency ω 0 of atoms defines the natural length scale 1/k 0 = c/ω 0 , where c is the vacuum speed of light. The ground state |g i of an isolated atom i is nondegenerate with the total angular momentum J g = 0, whereas the excited states |e i is three-fo...

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

Abstract: 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.

Cited by 66 publications

References 45 publications

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

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

Spectral backbone of excitation transport in ultracold Rydberg gases

Magnetic-Field-Driven Localization of Light in a Cold-Atom Gas

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