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We study out-of-time-ordered correlation (OTOC) functions in various randomquantum circuits and show that the average dynamics is governed by a Markovianpropagator. This is then used to study relaxation of OTOC to its long-timeaverage value in circuits with random single-qubit unitaries, finding thatrelaxation in general proceeds in two steps: in the first phase that lasts uptoan extensively long time the relaxation rate is given by a phantom eigenvalueof a non-symmetric propagator, whereas in the second phase the rate isdetermined by the true 2nd largest propagator eigenvalue. We also obtain exactOTOC dynamics on the light-cone and an expression for the average OTOC infinite random circuits with random two-qubit gates.

Two-step phantom relaxation of out-of-time-ordered correlations in random circuits