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