# A dynamical version of the SYK Model and the q-Brownian Motion

## TL;DR

This paper extends the SYK model to a multivariate, dynamical setting, connecting it with q-Brownian motion and analyzing fluctuations and higher correlations, revealing parallels with higher order freeness in random matrices.

## Contribution

It introduces a multivariate dynamical SYK model and establishes its relation to q-Brownian motion, extending fluctuation and correlation results.

## Key findings

- Eigenvalue distribution limits for multivariate SYK
- Connection between dynamical SYK and q-Brownian motion
- Higher order correlation function analysis

## 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 q-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.