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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In the physics literature, the random variables JX are typically taken to be Gaussian. We expect that a very similar result to ours will hold in this case as well, but we found the combinatorial problem discussed in Sec. IV A to be a bit more elegant for Rademacher random variables.
The authors of Ref. 14 called this the operator wave function.
However, such a seemingly optimal quantum walk likely does not maximize the probability of large size operators and perhaps does not even optimize the time-dependent average size.
