  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 $N\rightarrow\infty$. 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 $1/N$ and $1/q$, 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.
