  We construct a new and efficient cluster algorithm for updating strongly
coupled U(N) lattice gauge theories with staggered fermions in the chiral
limit. The algorithm uses the constrained monomer-dimer representation of the
theory and should also be of interest to researchers working on other models
with similar constraints. Using the new algorithm we address questions related
to the chiral limit of strongly coupled U(N) gauge theories beyond the mean
field approximation. We show that the infinite volume chiral condensate is
non-zero in three and four dimensions. However, on a square lattice of size $L$
we find $\sum_x < \bar\psi\psi(x) \bar\psi\psi(0) > \sim L^{2-\eta}$ for large
$L$ where $\eta = 0.420(3)/N + 0.078(4)/N^2$. These results differ from an
earlier conclusion obtained using a different algorithm. Here we argue that the
earlier calculations were misleading due to uncontrolled autocorrelation times
encountered by the previous algorithm.
