Unifying model for random matrix theory in arbitrary space dimensions

Giovanni M. Cicuta, Johannes Krausser, Rico Milkus, and Alessio Zaccone
Phys. Rev. E 97, 032113 – Published 14 March 2018

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, and 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, which clarifies the physical meaning of space dimension in these models. We feel that the approximate results for d=3 provided by our method may 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.

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  • Received 25 September 2017

DOI:https://doi.org/10.1103/PhysRevE.97.032113

©2018 American Physical Society

Physics Subject Headings (PhySH)

General Physics

Authors & Affiliations

Giovanni M. Cicuta1,*, Johannes Krausser2, Rico Milkus2, and Alessio Zaccone3,†

  • 1Dipartimento di Fisica, Università di Parma, Parco Area delle Scienze 7A, 43100 Parma, Italy
  • 2Statistical Physics Group, Department of Chemical Engineering and Biotechnology, University of Cambridge, Cambridge, CB3 0AS, United Kingdom
  • 3Statistical Physics Group, Department of Chemical Engineering and Biotechnology, and Cavendish Laboratory, University of Cambridge, Cambridge, CB3 0AS, United Kingdom

  • *cicuta@fis.unipr.it
  • az302@cam.ac.uk

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Issue

Vol. 97, Iss. 3 — March 2018

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