  We present results from numerical studies of the finite temperature phase
transition of the $(3+1)d$ O(N)-symmetric non-linear sigma model for $N=1,2$
and 3. We study the dependence of the width of the 3d critical region on $N$
and we show that the broken phase scaling region is much wider for N=2 and 3
than for N=1. We also compare the widths of the critical region in the low $T$
and high $T$ phases of the O(2) model and we show that the scaling region in
the broken phase is much wider than in the symmetric phase. We also report
results for the width of the scaling regions in the low $T$ phase$ (2+1)d$
Ising model and we show that the spatial correlation length has to be
approximately twice the lattice temporal extent before the 2d scaling region is
reached.
