Consider a Haar random quantum state of depth $d$. Consider any bipartition of this state. According to this paper (page $2$):
Haar-random states on $n$ qudits are nearly maximally entangled across all cuts simultaneously. Random quantum circuits on $L \times L \times \cdots L$ arrays of qudits achieve similar near-maximal entanglement across all possible cuts once the depth is $\Omega(L)$, and before this time, the entanglement often spreads “ballistically."
The comment indicates that the entanglement entropy is near maximal for linear depths. But, how much entanglement entropy is there for logarithmic and constant depths, and what is meant by "ballistic spreading"?