We compute the phase diagram in the N → ∞ limit for lattice RPN−1, CP and QP σ-models with the quartic action, and more generally for mixed isovector/isotensor models. We show that the N = ∞ limit exhibits phase transitions that are forbidden for any finite N . We clarify the origin of these pathologies by examining the exact solution of the one-dimensional model: we find that there are complex...