Operator dynamics in a Brownian quantum circuit

We view the operator spreading in chaotic evolution as a stochastic process of height growth. The height of an operator represents the size of its support and chaotic evolution increases the height. We consider N-spin models with all two-body interactions and embody the height picture in a random model. The exact solution shows that the mean height, being proportional to the squared commutator, grows exponentially within log N scrambling time and saturates in a manner of logistic function. We propose that the temperature dependence of the chaos bound could be due to initial height biased toward high operators, which has a smaller Lyapunov exponent.