  We apply the event-chain algorithm proposed by Bernard, Krauth and Wilson in
2009 to toy models of lattice QCD. We give a formal prove of stability of the
algorithm. We study its performance at the example of the massive Gaussian
model on the square and the simple cubic lattice, the $O(3)$-invariant
non-linear $\sigma$-model and the $SU(3) \times SU(3)$ principle chiral model
on the square lattice. In all these cases we find that critical slowing down is
essentially eliminated.
