The spin system analogues of recent studies of the string tension and Λ parameters of SU( N) gauge theories in 4 dimensions are carried out for the SU( N)×SU( N) and O( N) models in 2 dimensions. The relations between the Λ parameters of both the euclidean and hamiltonian formulation of the lattice models and the Λ parameter of the continuum models are obtained. We calculate the one-loop finite renormalization of the speed of light in the lattice hamiltonian formulationsof the O( N) and SU( N)×SU( N) models. Strong coupling calculations of the mass gaps of these spin models are done for all N and the constants of proportionality between the gap and the Λ parameter of the continuum models are obtained. These results are contrasted with similar calculations for the SU( N) gauge models in 3+1 dimensions. Identifying suitable coupling constants for discussing the N → ∞ limits, our numerical results suggest that the crossover from weak to strong coupling in the lattice O( N) models becomes less abrupt as N increases while the crossover for the SU( N)×SU( N) models becomes more abrupt. The crossover in SU( N) gauge theories also becomes more abrupt with increasing N, however, at an even greater rate than in the SU( N)×SU( N) spin models.