# Vibrational spectrum of Granular packings with random matrices

## TL;DR

This paper shows that the vibrational spectrum of granular materials can be modeled using random matrix theory, specifically the Laguerre orthogonal ensemble.

## Contribution

The paper introduces a random lattice model that captures the vibrational properties of granular matter and validates the use of the Laguerre orthogonal ensemble.

## Key findings

- The autocorrelation of the vibrational density of states matches the Laguerre orthogonal ensemble predictions.
- The Gaussian orthogonal ensemble fails to reproduce the statistical properties of jammed granular matter.
- A random lattice model successfully replicates the vibrational features of frictionless granular matter.

## Abstract

The vibrational spectrum of granular packings can be used as a signature of the jamming transition, with the density of states at zero frequency becoming nonzero at the transition. It has been proposed previously that the vibrational spectrum of granular packings can be approximately obtained from random matrix theory. Here, we show that the autocorrelation function of the density of states shows good agreement between dynamical numerical simulations of frictionless bead packs near the jamming point and the analytic predictions of the Laguerre orthogonal ensemble of random matrices; there is clear disagreement with the Gaussian orthogonal ensemble, establishing that the Laguerre ensemble correctly reproduces the universal statistical properties of jammed granular matter and excluding the Gaussian orthogonal ensemble. We also present a random lattice model which is a physically motivated variant of the random matrix ensemble. Numerical calculations reveal that this model reproduces the known features of the vibrational density of states of frictionless granular matter, while also retaining the correlation structure seen in the Laguerre random matrix theory.