Operator dynamics in Brownian quantum circuit

We view the operator spreading in chaotic evolution as a stochastic process of height growth. The height of an operator represents its spatial extent and a master equation governs the transition to higher operators. We derive and solve a master equation in a random N -spin model with all 2-body interactions. The mean height, being proportional to the squared commutator, will grow exponentially within log N scrambling time and saturates in a manner of logistic function. We propose that the chaos bound at finite temperature could be due to initial height biased towards the high operators, which has smaller Lyapunov exponent.