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Abstract In this note we discuss the phase space of the O(2N) vector model in the presence of a quadratic and a quartic interaction by writing the large-N effective potential using large charge methods in dimensions 2 < D < 4 and 4 < D < 6. Based on a simple discussion of the convexity properties of the grand potential, we find very different behavior in the two regimes: while in 2 < D < 4, the theory is well-behaved, the model in 4 < D < 6 leads to a complex CFT in the UV, consistently with earlier results. We also find a new metastable massive phase in the high-energy regime for the theory on the cylinder.