  We study the entanglement entropy of random partitions in one- and
two-dimensional critical fermionic systems. In an infinite system we consider a
finite, connected (hypercubic) domain of linear extent $L$, the points of which
with probability $p$ belong to the subsystem. The leading contribution to the
average entanglement entropy is found to scale with the volume as $a(p) L^D$,
where $a(p)$ is a non-universal function, to which there is a logarithmic
correction term, $b(p)L^{D-1}\ln L$. In $1D$ the prefactor is given by
$b(p)=\frac{c}{3} f(p)$, where $c$ is the central charge of the model and
$f(p)$ is a universal function. In $2D$ the prefactor has a different
functional form of $p$ below and above the percolation threshold.
