# Hamiltonian Dynamics of a Sum of Interacting Random Matrices

## TL;DR

This paper analytically investigates the dynamics of observables in a quantum system modeled by two interacting random matrices, revealing how non-relaxing components are determined by the Hamiltonian's spectrum and affecting late-time correlators.

## Contribution

It provides an exact analytical framework for understanding non-relaxing components in quantum observables using random matrix models and diagrammatic resummation techniques.

## Key findings

- Exact late-time correlator values for Hamiltonian H=A+B
- Power-law relaxation governed by H's spectrum, independent of A
- Modifications in OTOCs due to non-relaxing parts of A

## Abstract

In ergodic quantum systems, physical observables have a non-relaxing component if they "overlap" with a conserved quantity. In interacting microscopic models, how to isolate the non-relaxing component is unclear. We compute exact dynamical correlators governed by a Hamiltonian composed of two large interacting random matrices, $H=A+B$. We analytically obtain the late-time value of $\langle A(t) A(0) \rangle$; this quantifies the non-relaxing part of the observable $A$. The relaxation to this value is governed by a power-law determined by the spectrum of the Hamiltonian $H$, independent of the observable $A$. For Gaussian matrices, we further compute out-of-time-ordered-correlators (OTOCs) and find that the existence of a non-relaxing part of $A$ leads to modifications of the late time values and exponents. Our results follow from exact resummation of a diagrammatic expansion and hyperoperator techniques.