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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Research Article| August 05 2020
Non-perturbative dynamics of the operator size distribution in the Sachdev–Ye–Kitaev model
Andrew Lucas; Non-perturbative dynamics of the operator size distribution in the Sachdev–Ye–Kitaev model. J. Math. Phys. 1 August 2020; 61 (8): 081901. https://doi.org/10.1063/1.5133964
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