Operator growth in the SYK model

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.