Perturbation theory predictions and Monte Carlo simulations for the 2-d O(n) non-linear sigma-model

By using the results of a high-statistics (O(10^7) measurements) Monte Carlo simulation we test several predictions of perturbation theory on the O(n) non-linear sigma-model in 2 dimensions. We study the O(3) and O(8) models on large enough lattices to have a good control on finite-size effects. The magnetic susceptibility and three different definitions of the correlation length are measured. We check our results with large-n expansions as well as with standard formulae for asymptotic freedom up to 4 loops in the standard and effective schemes. For this purpose the weak coupling expansions of the energy up to 4 loops for the standard action and up to 3 loops for the Symanzik action are calculated. For the O(3) model we have used two different effective schemes and checked that they lead to compatible results. A great improvement in the results is obtained by using the effective scheme based on the energy at 3 and 4 loops. We find that the O(8) model follows very nicely (within few per mille) the perturbative predictions. For the O(3) model an acceptable agreement (within few per cent) is found.Comment: latex source + 15 e-postscript figures. It generates 26 pgs. Replaced version containing more corrections to scaling for the Symanzik action, more detailed explanation of the calculation of $C_\chi$ and a few more citation