Dissipation in the $1/D$ expansion for planar matrix models

TL;DR

This paper investigates the thermal properties of large matrix models in the planar limit using a 1/D expansion, revealing how energy degeneracies are lifted and dissipation timescales scale with D.

Contribution

It introduces a 1/D expansion approach to analyze thermal behavior and dissipation in planar matrix models with two quartic couplings, providing explicit calculations of correlators.

Findings

Degeneracy at large D is lifted at order 1/D.

Energy levels split by ~1/√D, indicating dissipation timescale ~√D.

High-temperature dissipation dominated by one quartic coupling.

Abstract

We consider the thermal behavior of a large number of matrix degrees of freedom in the planar limit. We work in $0+1$ dimensions, with $D$ matrices, and use $1/D$ as an expansion parameter. This can be thought of as a non-commutative large-$D$ vector model, with two independent quartic couplings for the two different orderings of the matrices. We compute a thermal two-point correlator to ${\cal O}(1/D)$ and find that the degeneracy present at large $D$ is lifted, with energy levels split by an amount $\sim 1/\sqrt{D}$. This implies a timescale for thermal dissipation $\sim \sqrt{D}$. At high temperatures dissipation is predominantly due to one of the two quartic couplings.