# Spectral Form Factor in Non-Gaussian Random Matrix Theories

## TL;DR

This paper analyzes the spectral form factor in non-Gaussian random matrix models with rich phase structures, revealing multi-critical behavior and universality at large times, with implications for chaotic quantum systems.

## Contribution

It provides explicit calculations of the spectral form factor in non-Gaussian matrix models, highlighting multi-criticality and universality in their spectral statistics.

## Key findings

- Spectral form factor exhibits critical behavior at phase transition points.
- Late time spectral correlations show universality across polynomial potentials.
- Estimated dip-time varies with potential, indicating phase-dependent dynamics.

## Abstract

We consider Random Matrix Theories with non-Gaussian potentials that have a rich phase structure in the large $N$ limit. We calculate the Spectral Form Factor (SFF) in such models and present them as interesting examples of dynamical models that display multi-criticality at short time-scales and universality at large time scales. The models with quartic and sextic potentials are explicitly worked out. The disconnected part of the Spectral Form Factor (SFF) shows a change in its decay behavior exactly at the critical points of each model. The dip-time of the SFF is estimated in each of these models. The late time behavior of all polynomial potential matrix models is shown to display a certain universality. This is related to the universality in the short distance correlations of the mean-level densities. We speculate on the implications of such universality for chaotic quantum systems including the SYK model.