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

TL;DR

The paper develops a dynamical, multivariate extension of the SYK model and proves its large-$n$ limit converges to a multivariate $q$-Gaussian family, with the dynamical version converging to the $q$-Brownian motion $S_q(t)$. It provides a detailed cumulant-based analysis of fluctuations and higher-order correlations, revealing a partitioned-permutation structure that mirrors higher-order freeness concepts in sparse random matrices. The work integrates non-commutative probability, $q$-Gaussian theory, and random matrix techniques to connect dynamical SYK dynamics with non-commutative stochastic processes, while highlighting analytic challenges in the multivariate $q$-Gaussian description. These results advance understanding of how sparse random matrices relate to $q$-deformed processes and suggest avenues for a richer theory of multivariate, dynamical freeness.

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.

Theorems & Definitions (12)

• Lemma 2.1
• Theorem 3.1
• Corollary 3.2
• Lemma 3.3
• Lemma 3.4
• proof : Proof of Theorem \ref{['MainTheorem']}
• Theorem 4.1
• proof
• Remark 4.2
• proof : Proof of Lemma \ref{['Lemma0']}