Asymptotic Scaling in the Two-Dimensional O(3) σ Model at Correlation Length 105

We carry out a high-precision Monte Carlo simulation of the two-dimensional O(3)-invariant σ model at correlation lengths ξ up to $\sim$1$0^5$. Our work employs a 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% at $\xi \sim {10}^2$, decreases to $\approx$4% at $\xi \sim {10}^5$.