We carry out a high-precision Monte Carlo simulation of the two-dimensional O (3)-invariant
σ model at correlation lengths ξ up to∼ 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≈ 25% at ξ∼ 10 2, decreases to≈ 4% at ξ∼ 10 5.

We carry out a high-precision simulation of the two-dimensional SU (3) principal chiral
model at correlation lengths ξ 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 ξ≳ 10 3 we observe good asymptotic scaling in the bare coupling; at ξ≈ 4× …