Taming NSPT fluctuations in $O(N)$ Non-Linear Sigma Model: simulations in the large $N$ regime

The Non-Linear Sigma Model (NLSM) is an example of a field theory on a target space exhibiting intricate geometry. One remarkable characteristic of the NLSM is asymptotic freedom, which triggers interest in perturbative calculations. In the lattice formulation of NLSM, one would naturally rely on Numerical Stochastic Perturbation Theory (NSPT) to conduct high-order computations. However, when dealing with low-dimensional systems, NSPT reveals increasing statistical fluctuations with higher and higher orders. This of course does not come as a surprise and one is ready to live with this, as long as the noise is not going to completely kill the signal, which unfortunately in some models does take place. We investigate how, in the $O(N)$ context, this behaviour strongly depends on $N$. As expected, larger $N$ values make higher-order computations feasible.