Asymptotic scaling in the two-dimensional $SU(3)$ $\sigma$-model at correlation length $4\times {10}^5$

We carry out a high-precision simulation of the two-dimensional $SU(3)$ principal chiral model at correlation lengths $\xi$ up to $\approx! 4 \times 10^5$, using a multi-grid 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 $\xi \text{gtapprox}{10}^3$ 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.