Asymptotic scaling in the two-dimensional SU(3)σmodel at correlation length 4 ×105

We carry out a high-precision simulation of the two-dimensional SU(3) principal chiral model at correlation lengths $\xi$ up to ≈4 × ${10}^5$, using a multigrid Monte Carlo (MGMC) algorithm. We extrapolate the finite-volume Monte Carlo data to infinite volume using finite-size-scaling theory, and we discuss carefully the systematic and statistical errors in this extrapolation. We then compare the extrapolated data to the renormalization-group predictions. For $\xi \gtrsim {10}^3$ we observe good asymptotic scaling in the bare coupling; at $\xi \approx 4\times {10}^5$ the nonperturbative constant is within 2-3% of its predicted limiting value.