  The large $N$ expansion plays a fundamental role in quantum and statistical
field theory. We show on the example of the O$(N)$ model that at $N=\infty$,
its traditional implementation misses in all dimensions below four some fixed
points of the renormalization group. These new fixed points show singularities
at $N=\infty$ in their effective potential that become a boundary layer at
finite $N$. We show that they have a physical impact on the multicritcal
physics of the $O(N$) model at finite $N$. We also show that the mechanism at
play holds also for the O($N$)$\otimes$O(2) model and is thus probably generic.
