  We explore the operator dynamics in a random $N$-spin model with pairwise
interactions (Brownian quanum circuit). We introduce the height $h$ of an
operator to characterize its spatial extent, and derive the master equation of
the height probability distribution. The study of an initial simple operator
with $h = 1$ (minimal nonzero height) shows that the mean height, which is
proportional to the squared commutator, has an initial exponential growth. It
then slows down around the scrambling time $\sim\log N$ and finally saturates
to a steady state in a manner similar to the logistic function. The deviation
to the logistic function is due to the large fluctuations (order $N$) in the
intermediate time. Moreover, we find that the exponential growth rate (quantum
Lyapunov exponent) is smaller for initial operator with $\langle h\rangle\gg
1$. Based on this observation, we propose that the chaos bound at finite
temperature can be produced by an initial operator whose height distribution is
biased towards higher operators. We numerically test the power law initial
height distribution $1/h^{\alpha} $ in a Brownian circuit with number of spin
$N=10000$ and show that the Lyapunov exponent is linearly constrained by
$\alpha$ before reaching the infinite temperature value.
