Large fluctuations in NSPT computations: a lesson from $O(N)$ non-linear sigma models

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: for a large number of internal degrees of freedom (i.e. for large enough $N$), NSPT perturbative computation can be pushed to large perturbative orders $n$. By re-expressing our perturbative expansions as power series in the $gN$ ('t Hooft) coupling, we show some evidence that at any given order $n$ there is a tendency to gaussianity for the stochastic process distributions at large $N$. By summing our series, we can verify leading order results for the energy and its (field theoretic) variance in the large $N$ limit. We finally establish general relationships between the various perturbative orders in the expansion of the (field theoretic) variance of a given observable and combinations of variances and covariances of given orders NSPT stochastic processes. Having established all this, we conclude discussing interesting applications of NSPT computations in the context of theories similar to $O(N)$ (i.e. $CP(N-1)$ models).