  A theory of the measurement-induced entanglement phase transition for
free-fermion models in $d>1$ dimensions is developed. The critical point
separates a gapless phase with $\ell^{d-1} \ln \ell$ scaling of the second
cumulant of the particle number and of the entanglement entropy and an area-law
phase with $\ell^{d-1}$ scaling, where $\ell$ is a size of the subsystem. The
problem is mapped onto an SU($R$) replica non-linear sigma model in $d+1$
dimensions, with $R\to 1$. Using renormalization-group analysis, we calculate
critical indices in one-loop approximation justified for $d = 1+ \epsilon$ with
$\epsilon \ll 1$. Further, we carry out a numerical study of the transition for
a $d=2$ model on a square lattice, determine numerically the critical point,
and estimate the critical index of the correlation length, $\nu \approx 1.4$.
