Operator growth bounds in a cartoon matrix model

TL;DR

This paper investigates operator growth in a Majorana fermion model on a complete graph, demonstrating that the scrambling time scales at least logarithmically with system size, aligning with the fast scrambling conjecture.

Contribution

The paper proves, non-perturbatively and without ensemble averaging, that the scrambling time in this matrix-like model is at least logarithmic in system size, connecting it to holographic duality.

Findings

Scrambling time is at least of order log N.

Model exhibits features similar to matrix models and holography.

Operator growth dynamics are analyzed without ensemble averaging.

Abstract

We study operator growth in a model of $N(N-1)/2$ interacting Majorana fermions, which live on the edges of a complete graph of $N$ vertices. Terms in the Hamiltonian are proportional to the product of $q$ fermions which live on the edges of cycles of length $q$. This model is a cartoon "matrix model": the interaction graph mimics that of a single-trace matrix model, which can be holographically dual to quantum gravity. We prove (non-perturbatively in $1/N$, and without averaging over any ensemble) that the scrambling time of this model is at least of order $\log N$, consistent with the fast scrambling conjecture. We comment on apparent similarities and differences between operator growth in our "matrix model" and in the melonic models.