A unifying model for random matrix theory in arbitrary space dimensions

Abstract

A sparse random block matrix model suggested by the Hessian matrix used in the study of elastic vibrational modes of amorphous solids is presented and analyzed. By evaluating some moments, benchmarked against numerics, differences in the eigenvalue spectrum of this model in different limits of space dimension $d$, and for arbitrary values of the lattice coordination number $Z$, are shown and discussed. As a function of these two parameters (and their ratio $Z/d$), the most studied models in random matrix theory (Erdos-Renyi graphs, effective medium, replicas) can be reproduced in the various limits of block dimensionality $d$. Remarkably, the Marchenko-Pastur spectral density (which is recovered by replica calculations for the Laplacian matrix) is reproduced exactly in the limit of infinite size of the blocks, or $d\to \infty$, which for the first time clarifies the physical meaning of space dimension in these models. The approximate results for $d=3$ provided by our method have many potential applications in the future, from the vibrational spectrum of glasses and elastic networks, to wave-localization, disordered conductors, random resistor networks and random walks.

References in corpus (7)

• Local inversion-symmetry breaking controls the boson peak in glasses and crystals
• Cavity Approach to the Spectral Density of Sparse Symmetric Random Matrices
• Spectra of Sparse Random Matrices
• On the localization transition in symmetric random matrices
• Finitely coordinated models for low-temperature phases of amorphous systems
• Equitable random graphs
• Equivalence of replica and cavity methods for computing spectra of sparse random matrices

Cited by in corpus (16)

• Relaxation and vibrational properties in metal alloys and other disordered systems
• Mean-field model for the density of states of jammed soft spheres
• A simple random matrix model for the vibrational spectrum of jammed packings
• Vibrational density of states and specific heat in glasses from random matrix theory
• Random matrices with row constraints and eigenvalue distributions of graph Laplacians
• Complete mathematical theory of the jamming transition: A perspective
• Vibrational density of states of jammed packing: mean-field theory
• Dynamics of Disordered Mechanical Systems with Large Connectivity, Free Probability Theory, and Quasi-Hermitian Random Matrices
• The Spectral Boundary of Block Structured Random Matrices
• Proof of a conjecture on the infinite dimension limit of a unifying model for random matrix theory
• Random matrix approach to plasmon resonances in the random impedance network model of disordered nanocomposites
• Hammerstein equations for sparse random matrices
• Sparse Random Block Matrices
• Noncrossing partition flow and random matrix models
• Statistical mechanics of vector Hopfield network near and above saturation
• Mean-field density of states of a small-world model and a jammed soft spheres model