  The phase transition temperature for the Bose-Einstein condensation (BEC) of
weakly-interacting Bose gases in three dimensions is known to be related to
certain non-universal properties of the phase transition of three-dimensional
O(2) symmetric $\phi^4$ theory. These properties have been measured previously
in Monte Carlo lattice simulations. They have also been approximated
analytically, with moderate success, by large $N$ approximations to O($N$)
symmetric $\phi^4$ theory. To begin investigating the region of validity of the
large $N$ approximation in this application, I have applied the same Monte
Carlo technique developed for the O(2) model ([5]) to O(1) and O(4) theories.
My results indicate that there might exist some theoretically unanticipated
systematic errors in the extrapolation of the continuum value from lattice
Monte Carlo results. The final results show that the difference between
simulations and NLO large $N$ calculations does not improve significantly from
N=2 to N=4. This suggests one would need to simulate yet larger $N$'s to see
true large $N$ scaling of the difference. Quite unexpectedly (and presumably
accidentally), my Monte Carlo result for N=1 seems to give the best agreement
with the large $N$ approximation among the three cases.
