Two- and three-point functions at criticality: Monte Carlo simulations of the improved three-dimensional Blume-Capel model

We compute two- and three-point functions at criticality for the three-dimensional Ising universality class. To this end we simulate the improved Blume-Capel model at the critical temperature on lattices of a linear size up to $L=1600$. As check also simulations of the spin-1/2 Ising model are performed. We find $f_{\sigma \sigma ϵ}=1.051(1)$ and $f_{\epsilon \epsilon \epsilon} =1.533(5)$ for operator product expansion coefficients. These results are consistent with but less precise than those recently obtained by using the bootstrap method. An important ingredient in our simulations is a variance reduced estimator of $N$-point functions. Finite size corrections vanish with $L^{-{\mathrm{\Delta }}_{ϵ}}$, where $L$ is the linear size of the lattice and ${\mathrm{\Delta }}_{ϵ}$ is the scaling dimension of the leading $Z_2$-even scalar $ϵ$.