  In the last three decades, Numerical Stochastic Perturbation Theory (NSPT)
has proven to be an excellent tool for calculating perturbative expansions in
theories such as Lattice QCD, for which standard, diagrammatic perturbation
theory is known to be cumbersome. Despite the significant success of this
stochastic method and the improvements made in recent years, NSPT apparently
cannot be successfully implemented in low-dimensional models due to the
emergence of huge statistical fluctuations: as the perturbative order gets
higher, the signal to noise ratio is simply not good enough. This does not come
as a surprise, but on very general grounds, one would expect that the larger
the number of degrees of freedom, the less severe the fluctuations will be. By
simulating $2D$ $O(N)$ non-linear sigma models for different values of $N$, we
show that indeed the fluctuations are tamed in the large $N$ limit, meeting our
expectations. Having established this, we conclude discussing interesting
applications of NSPT in the context of these theories.
