Non-perturbative dynamics of the operator size distribution in the Sachdev-Ye-Kitaev model

We prove non-perturbative bounds on the time evolution of the probability distribution of operator size in the $q$-local Sachdev-Ye-Kitaev model with $N$ fermions, for any even integer $q>2$ and any positive even integer $N>2q$. If the couplings in the Hamiltonian are independent and identically distributed Rademacher random variables, the infinite temperature many-body Lyapunov exponent is almost surely finite as $N\to \mathrm{\infty }$. In the limit $q \rightarrow \infty$,$N\rightarrow \infty$,$q^{6+\delta}/N \rightarrow 0$, the shape of the size distribution of a growing fermion, obtained by leading order perturbation calculations in $1\mathrm{/}N$ and $1\mathrm{/}q$, is similar to a distribution that locally saturates our constraints. Our proof is not based on Feynman diagram resummation; instead, we note that the operator size distribution obeys a continuous time quantum walk with bounded transition rates, to which we apply concentration bounds from classical probability theory.