Nonuniformity 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 nonuniform 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 nonuniformity.