A single defect approximation for localized states on random lattices

Abstract

Geometrical disorder is present in many physical situations giving rise to eigenvalue problems. The simplest case of diffusion on a random lattice with fluctuating site connectivities is studied analytically and by exact numerical diagonalizations. Localization of eigenmodes is shown to be induced by geometrical defects, that is sites with abnormally low or large connectivities. We expose a ``single defect approximation'' (SDA) scheme founded on this mechanism that provides an accurate quantitative description of both extended and localized regions of the spectrum. We then present a systematic diagrammatic expansion allowing to use SDA for finite-dimensional problems, e.g. to determine the localized harmonic modes of amorphous media.

final version as published, 6 pages, 1 ps-figure

References in corpus (2)

• Thermodynamics of glasses: a first principle computation
• Optimisation problems and replica symmetry breaking in finite connectivity spin-glasses

Cited by in corpus (75)

• Sparse random matrices: the eigenvalue spectrum revisited
• Cavity Approach to the Spectral Density of Sparse Symmetric Random Matrices
• The Shannon and the Von Neumann entropy of random networks with heterogeneous expected degree
• Spectra of Sparse Random Matrices
• Laplacian Spectra as a Diagnostic Tool for Network Structure and Dynamics
• Random incidence matrices: moments of the spectral density
• Vibrational spectrum of topologically disordered systems
• Cavity approach to the spectral density of non-Hermitian sparse matrices
• Theoretical study of collective modes in DNA at ambient temperature
• On the localization transition in symmetric random matrices
• May's Instability in Large Economies
• Cavity and replica methods for the spectral density of sparse symmetric random matrices
• Spectra of random networks with arbitrary degrees
• Mean-field model for the density of states of jammed soft spheres
• Stress Relaxation of Near-Critical Gels
• Spectral properties of the Laplacian on bond-percolation graphs
• On the stochastic dynamics of disordered spin models
• The dynamical structure factor in topologically disordered systems
• Localization of eigenvectors in random graphs
• Spectra of Modular and Small-World Matrices
• Cavity approach to the first eigenvalue problem in a family of symmetric random sparse matrices
• Lifshitz tails for spectra of Erdős--Rényi random graphs
• The dynamical structure factor in disordered systems
• Spectral density of random graphs with topological constraints
• Spectral Statistics of Instantaneous Normal Modes in Liquids and Random Matrices
• Revealing the Micro-Structure of the Giant Component in Random Graph Ensembles
• Limitations in the spectral method for graph partitioning: detectability threshold and localization of eigenvectors
• First eigenvalue/eigenvector in sparse random symmetric matrices: influences of degree fluctuation
• Spectral Density of Sparse Sample Covariance Matrices
• Anderson Localization in Euclidean Random Matrices
• Critical Dynamics of Gelation
• Heterogeneous micro-structure of percolation in sparse networks
• Dynamics of dilute disordered models: a solvable case
• Vector vibrations and the Ioffe-Regel crossover in disordered lattices
• Universal hypotrochoidic law for random matrices with cyclic correlations
• A unifying model for random matrix theory in arbitrary space dimensions
• Equivalence of replica and cavity methods for computing spectra of sparse random matrices
• On the high-density expansion for Euclidean Random Matrices
• Eigenvalue spectra and stability of directed complex networks
• Spectral Density of Complex Networks with a Finite Mean Degree
• Return probabilities and hitting times of random walks on sparse Erdos-Renyi graphs
• Fully localized and partially delocalized states in the tails of Erdös-Rényi graphs in the critical regime
• Lifshitz tails on the Bethe lattice: a combinatorial approach
• Nature of vibrational eigenmodes in topologically disordered solids
• Multifractal phase in the weighted adjacency matrices of random Erdös-Rényi graphs
• Random Vibrational Networks and Renormalization Group
• Second largest Eigenpair Statistics for Sparse Graphs
• Top Eigenpair Statistics for Weighted Sparse Graphs
• Glassy dynamics on networks: local spectra and return probabilities
• Spectral properties of the trap model on sparse networks
• Index statistical properties of sparse random graphs
• Disentangling Giant Component and Finite Cluster Contributions in Sparse Matrix Spectra
• Anomalous stress relaxation in random macromolecular networks
• Localization in random bipartite graphs: numerical and empirical study
• Calculation of mean spectral density for statistically uniform tree-like random models
• Multifractality and statistical localization in highly heterogeneous random networks
• The integrated density of states of the random graph Laplacian
• Asymptotic behavior of the density of states on a random lattice
• Persistent individual bias in a voter model with quenched disorder
• Spectral properties, localization transition and multifractal eigenvectors of the Laplacian on heterogeneous networks
• Localization transition in non-Hermitian systems depending on reciprocity and hopping asymmetry
• Stability and dynamical properties of material flow systems on random networks
• Robust extended states in Anderson model on partially disordered random regular graphs
• Random sparse generators of Markovian evolution and their spectral properties
• Linear processes in high-dimension: phase space and critical properties
• Random matrix approach to plasmon resonances in the random impedance network model of disordered nanocomposites
• Spectral properties of disordered fully-connected graphs
• Fat-tailed distribution derived from first eigenvector of symmetric random sparse matrix
• Eigenvalue spectral tails and localization properties of asymmetric networks
• Real symmetric random matrices and replicas
• A path integral approach to sparse non-Hermitian random matrices
• Top eigenpair statistics of diluted Wishart matrices
• Structure Learning in Inverse Ising Problems Using ${\ell }_2$-Regularized Linear Estimator
• Spectral properties of unimodular lattice triangulations
• Fragility of spectral clustering for networks with an overlapping structure