Asymptotic Scaling in the Two-Dimensional $O(3)$ $\sigma$-Model at Correlation Length ${10}^5$

We carry out a high-precision Monte Carlo simulation of the two-dimensional $O(3)$-invariant $\sigma$-model at correlation lengths $\xi$ up to $\sim {10}^5$. Our work employs a new and powerful method for extrapolating finite-volume Monte Carlo data to infinite volume, based on finite-size-scaling theory. We discuss carefully the systematic and statistical errors in this extrapolation. We then compare the extrapolated data to the renormalization-group predictions. The deviation from asymptotic scaling, which is $\approx 25\mathrm{\%}$ at $\xi \sim 10^2$,decreasesto$\approx 4%$at$\xi \sim 10^5$.