Operator growth bounds in a cartoon matrix model

We study operator growth in a model of N(N - 1)/2 interacting Majorana fermions that live on the edges of a complete graph of N vertices. Terms in the Hamiltonian are proportional to the product of q fermions that 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 logN, 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.