Non-uniformity of the 1/(N) expansion for two-dimensional O(N) models

We point out that the 1/ N expansion, which is widely invoked to infer properties of the 2D O( N) models, is non-uniform in the temperature, i.e. with decreasing temperature the 1/ N expansion truncated at a fixed order deviates more and more from the true answer. This fact precludes the use of the expansion to deduce low-temperature properties such as asymptotic scaling for those models. By contrast, in the 1D O( N) chains, there are no signs of such a non-uniformity.