Dynamics of disordered mechanical systems with large connectivity, free probability theory, and quasi-Hermitian random matrices

Joshua Feinberg, Roman Riser

Research output: Contribution to journalArticlepeer-review


Disordered mechanical systems with high connectivity represent a limit opposite to the more familiar case of disordered crystals. Individual ions in a crystal are subjected essentially to nearest-neighbor interactions. In contrast, the systems studied in this paper have all their degrees of freedom coupled to each other. Thus, the problem of linearized small oscillations of such systems involves two full positive-definite and non-commuting matrices, as opposed to the sparse matrices associated with disordered crystals. Consequently, the familiar methods for determining the averaged vibrational spectra of disordered crystals, introduced many years ago by Dyson and Schmidt, are inapplicable for highly connected disordered systems. In this paper we apply random matrix theory (RMT) to calculate the averaged vibrational spectra of such systems, in the limit of infinitely large system size. At the heart of our analysis lies a calculation of the average spectrum of the product of two positive definite random matrices by means of free probability theory techniques. We also show that this problem is intimately related with quasi-hermitian random matrix theory (QHRMT), which means that the ‘hamiltonian’ matrix is hermitian with respect to a non-trivial metric. This extends ordinary hermitian matrices, for which the metric is simply the unit matrix. The analytical results we obtain for the spectrum agree well with our numerical results. The latter also exhibit oscillations at the high-frequency band edge, which fit well the Airy kernel pattern. We also compute inverse participation ratios of the corresponding amplitude eigenvectors and demonstrate that they are all extended, in contrast with conventional disordered crystals. Finally, we compute the thermodynamic properties of the system from its spectrum of vibrations. In addition to matrix model analysis, we also study the vibrational spectra of various multi-segmented disordered pendula, as concrete realizations of highly connected mechanical systems. A universal feature of the density of vibration modes, common to both pendula and the matrix model, is that it tends to a non-zero constant at vanishing frequency.

Original languageEnglish
Article number168456
JournalAnnals of Physics
StatePublished - Dec 2021

Bibliographical note

Funding Information:
This research was supported by the Israel Science Foundation (ISF) under grant No. 2040/17 . Computations presented in this work were performed on the Hive computer cluster at the University of Haifa, which is partly funded by ISF, Israel grant 2155/15 .

Publisher Copyright:
© 2021 Elsevier Inc.


  • Free probability theory and S-transforms
  • Large connectivity
  • Quasi-Hermitian matrices
  • Random matrix theory
  • Spectral dimension
  • Vibrational spectrum of disordered systems

ASJC Scopus subject areas

  • Physics and Astronomy (all)


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