A dynamical version of the SYK model and the q-Brownian motion

Abstract

We extend recent results on the asymptotic eigenvalue distribution of the SYK model to the multivariate case and relate the limit of a dynamical version of the SYK model with the [Formula: see text]-Brownian motion, a non-commutative deformation of classical Brownian motion. Furthermore, we extend the results for fluctuations to the multivariate setting and treat also higher correlation functions. The structure of our results for the sparse SYK random matrices resembles the formulas for higher order freeness for ordinary GUE random matrices.