  We numerically study the measurement-driven quantum phase transition of
Haar-random quantum circuits in $1+1$ dimensions. By analyzing the tripartite
mutual information we are able to make a precise estimate of the critical
measurement rate $p_c = 0.17(1)$. We extract estimates for the associated bulk
critical exponents that are consistent with the values for percolation, as well
as those for stabilizer circuits, but differ from previous estimates for the
Haar-random case. Our estimates of the surface order parameter exponent appear
different from that for stabilizer circuits or percolation, but we are unable
to definitively rule out the scenario where all exponents in the three cases
match. Moreover, in the Haar case the prefactor for the entanglement entropies
$S_n$ depends strongly on the R\'enyi index $n$; for stabilizer circuits and
percolation this dependence is absent. Results on stabilizer circuits are used
to guide our study and identify measures with weak finite-size effects. We
discuss how our numerical estimates constrain theories of the transition.
