Possible third-order phase transition in the large-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>N</mml:mi></mml:math>lattice gauge theory

The large-$N$ limit of the two-dimensional $\mathrm{U}(N)$ (Wilson) lattice gauge theory is explicitly evaluated for all fixed $\ensuremath{\lambda}={g}^{2}N$ by steepest-descent methods. The $\ensuremath{\lambda}$ dependence is discussed and a third-order phase transition, at $\ensuremath{\lambda}=2$, is discovered. The possible existence of such a weak- to strong-coupling third-order phase transition in the large-$N$ four-dimensional lattice gauge theory is suggested, and its meaning and implications are discussed.