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Unifying model for random matrix theory in arbitrary space dimensions.

Giovanni M Cicuta1, Johannes Krausser2, Rico Milkus2

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A new sparse random block matrix model analyzes elastic vibrational modes in amorphous solids. This model reproduces established random matrix theories and clarifies the physical meaning of space dimension.

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Area of Science:

  • Condensed Matter Physics
  • Statistical Mechanics
  • Materials Science

Background:

  • Amorphous solids exhibit complex vibrational properties governed by their disordered structure.
  • Understanding these properties is crucial for applications in materials science and condensed matter physics.
  • Existing models in random matrix theory offer frameworks but may not fully capture specific aspects of disordered systems.

Purpose of the Study:

  • To introduce and analyze a novel sparse random block matrix model.
  • To investigate the eigenvalue spectrum of this model across different space dimensions (d) and lattice coordination numbers (Z).
  • To establish connections between this new model and existing random matrix theory frameworks.

Main Methods:

  • Development of a sparse random block matrix model inspired by Hessian matrices.
  • Evaluation of model moments and benchmarking against numerical simulations.
  • Analysis of the eigenvalue spectrum as a function of space dimension (d) and coordination number (Z).

Main Results:

  • The model successfully reproduces established random matrix theory models (Erdos-Renyi, effective medium, replicas) in various limits of block dimensionality.
  • The Marchenko-Pastur spectral density is exactly recovered in the limit of infinite block size (d→∞), clarifying the role of space dimension.
  • The study reveals differences in the eigenvalue spectrum based on the ratio Z/d.

Conclusions:

  • The proposed model offers a unified framework for understanding disordered systems.
  • It provides physical insights into the role of space dimension in random matrix models.
  • The model's approximate results for d=3 suggest potential applications in diverse fields like wave localization and disordered conductors.