  We investigate the critical behavior of the entanglement transition induced
by projective measurements in (Haar) random unitary quantum circuits. Using a
replica approach, we map the calculation of the entanglement entropies in such
circuits onto a two-dimensional statistical mechanics model. In this language,
the area- to volume-law entanglement transition can be interpreted as an
ordering transition in the statistical mechanics model. We derive the general
scaling properties of the entanglement entropies and mutual information near
the transition using conformal invariance. We analyze in detail the limit of
infinite on-site Hilbert space dimension in which the statistical mechanics
model maps onto percolation. In particular, we compute the exact value of the
universal coefficient of the logarithm of subsystem size in the $n$th R\'enyi
entropies for $n \geq 1$ in this limit using relatively recent results for
conformal field theory describing the critical theory of 2D percolation, and we
discuss how to access the generic transition at finite on-site Hilbert space
dimension from this limit, which is in a universality class different from 2D
percolation. We also comment on the relation to the entanglement transition in
Random Tensor Networks, studied previously in Ref. 1.
