  We study the probability distribution of entanglement in the Quantum
Symmetric Simple Exclusion Process, a model of fermions hopping with random
Brownian amplitudes between neighboring sites. We consider a protocol where the
system is initialized in a pure product state of $M$ particles, and focus on
the late-time distribution of R\'enyi-$q$ entropies for a subsystem of size
$\ell$. By means of a Coulomb gas approach from Random Matrix Theory, we
compute analytically the large-deviation function of the entropy in the
thermodynamic limit. For $q>1$, we show that, depending on the value of the
ratio $\ell/M$, the entropy distribution displays either two or three distinct
regimes, ranging from low- to high-entanglement. These are connected by points
where the probability density features singularities in its third derivative,
which can be understood in terms of a transition in the corresponding charge
density of the Coulomb gas. Our analytic results are supported by numerical
Monte Carlo simulations.
