  Phase transitions can occur in one-dimensional classical statistical
mechanics at non-zero temperature when the number of components N of the spin
is infinite. We show how to solve such magnets in one dimension for any N, and
how the phase transition develops at N = infinity. We discuss SU(N) and Sp(N)
magnets, where the transition is second-order. In the new high-temperature
phase, the correlation length is zero. We also show that for the SU(N) magnet
on exactly three sites with periodic boundary conditions, the transition
becomes first order.
