We study the large-$N$ reduced model recently proposed by the present authors. This model is a modified version of the Eguchi-Kawai model incorporating twisted boundary conditions. It is shown that the Schwinger-Dyson equations of our model are the same as in the infinite-lattice theory provided ${\mathrm{}\left[\mathrm{U}\right(1\mathrm{}\left)\right]}^4$ symmetry is not spontaneously broken. We study the model at strong coupling, weak coupling, and intermediate coupling using analytical and Monte Carlo techniques. At weak coupling, it is shown that for a particular choice of twist, ${\mathrm{}\left[\mathrm{U}\right(1\mathrm{}\left)\right]}^4$ symmetry is not broken and we prove how one recovers usual planar perturbation theory. Monte Carlo data for $\chi$ ratios show striking agreement with Wilson-theory results.

• Received 15 December 1982

DOI:https://doi.org/10.1103/PhysRevD.27.2397