Finite size effects, scaling and algorithms for lattice CPN-1

We present simple model predictions for the spectrum of CPN-1 in a periodic box and use them to interpret the strong finite size effects observed in lattice simulations at medium values of N. The asymptotic scaling behaviour of alternative lattice actions is explored. A detailed comparison confirms earlier suggestions that an action which explicitly includes the effects of the auxilliary gauge field has superior scaling properties. Variations of a recently introduced class of multigrid algorithms are tested. These show good decorrelation properties but do not appear to approach complete elimination of critical slowing, particularly at large N. (orig.)