Crossover from Weak to Strong Coupling inSU(N)Lattice Gauge Theories

Coupling constant renormalization for $\mathrm{SU}\mathrm{}\left(N\right)\mathrm{}$ lattice gauge theories is studied with Hamiltonian methods. Expansions to $O\left(g^{-16\mathrm{}}\right)$ suggest that the crossover from weak to strong coupling occurs progressively more abruptly in the variable ${\lambda }^2=g^{2}N$ as $N$ increases. The constants relating the string tension to the scale-breaking parameter $\Lambda$ are estimated for all $N$. It is suggested that the Callan-Symanzik $\beta$ function develops a kink in the $N\to \infty$ limit at a nonzero value of ${\lambda }^2$.