  We study strongly coupled lattice QCD with $N$ colors of staggered fermions
in 3+1 dimensions. While mean field theory describes the low temperature
behavior of this theory at large $N$, it fails in the scaling region close to
the finite temperature second order chiral phase transition. The universal
critical region close to the phase transition belongs to the 3d XY universality
class even when $N$ becomes large. This is in contrast to Gross-Neveu models
where the critical region shrinks as $N$ (the number of flavors) increases and
mean field theory is expected to describe the phase transition exactly in the
limit of infinite $N$. Our work demonstrates that close to second order phase
transitions infrared fluctuations can sometimes be important even when $N$ is
strictly infinite.
