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. In the limit q, N, and q6+δ/N → 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.

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