Operator growth in the SYK model

TL;DR

This paper investigates how operators grow in size over time in the SYK model, linking the growth to quantum chaos and providing numerical and analytical methods to characterize the distribution.

Contribution

It introduces a detailed analysis of operator size distribution in the SYK model, including numerical evaluation and large-q analytical calculations.

Findings

Operator size distribution shifts towards larger operators over time.

Initial growth rate is exponential, governed by the chaos exponent.

Explicit large-q expansion results for the size distribution.

Abstract

We discuss the probability distribution for the "size" of a time-evolving operator in the SYK model. Scrambling is related to the fact that as time passes, the distribution shifts towards larger operators. Initially, the rate is exponential and determined by the infinite-temperature chaos exponent. We evaluate the size distribution numerically for $N = 30$, and show how to compute it in the large-$N$ theory using the dressed fermion propagator. We then evaluate the distribution explicitly at leading nontrivial order in the large-$q$ expansion.