# Power-law out of time order correlation functions in the SYK model

## TL;DR

This paper analyzes the SYK model's finite temperature properties, revealing a universal power-law decay in out-of-time-order correlations at large times, derived through a Liouville quantum mechanics approach.

## Contribution

It introduces an exact integration method over conformal modes to study the SYK model, uncovering a universal power-law decay in correlation functions at large times.

## Key findings

- Out-of-time-order correlation functions exhibit a crossover from exponential decay to a $t^{-6}$ power-law.
- The analysis employs a mapping to Liouville quantum mechanics for exact results.
- The power-law behavior appears at time scales proportional to the number of particles.

## Abstract

We evaluate the finite temperature partition sum and correlation functions of the Sachdev-Ye-Kitaev (SYK) model. Starting from a recently proposed mapping of the SYK model onto Liouville quantum mechanics, we obtain our results by exact integration over conformal Goldstone modes reparameterizing physical time. Perhaps, the least expected result of our analysis is that at time scales proportional to the number of particles the out of time order correlation function crosses over from a regime of exponential decay to a universal $t^{-6}$ power-law behavior.