  We compute the phase diagram in the N\to\infty limit for lattice RP^{N-1},
CP^{N-1} and QP^{N-1} sigma-models with the quartic action, and more generally
for mixed isovector/isotensor models. We show that the N=\infty 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 zeros of the partition function that tend to the
real axis as N\to\infty. We conjecture the correct phase diagram for finite N
as a function of the spatial dimension d. Along the way, we prove some new
correlation inequalities for a class of N-component sigma-models, and we obtain
some new results concerning the complex zeros of confluent hypergeometric
functions.
