Two-dimensionalSU(N)×SU(N)chiral models on the lattice

Lattice $\mathrm{SU}\mathrm{}\left(N\right)\mathrm{}\times \mathrm{SU}\mathrm{}\left(N\right)\mathrm{}$ chiral models are analyzed by strong and weak coupling expansions and by numerical simulations. Twelfth-order strong coupling series for the free and internal energy are obtained for all $N\ge 6$. Three-loop contributions to the internal energy and to the lattice $\beta$ function are evaluated for all $N$ and nonuniversal corrections to the asymptotic $\Lambda$ parameter are computed in the "temperature" and the "energy" schemes. Numerical simulations confirm a faster approach to asymptopia of the energy scheme. A phenomenological correlation between the peak in the specific heat and the dip of the $\beta$ function is observed. Tests of scaling are performed for various physical quantities, finding substantial scaling at $\xi \gtrsim 2$. In particular, at $N=6\mathrm{}$ three different mass ratios are determined numerically and found in agreement, within statistical errors of about 1%, with the theoretical predictions from the exact $S$-matrix theory.