Figure 2.

Variance in the estimator of the mean magnetization as a function of the simulation length t_sim measured in units of MC passes. The symbols show the results of a regular MC simulation (top, inverted triangles) and ST simulations with N=3, 5, 7, and 9 temperatures (top to bottom) and corresponding upper temperatures T_N=2.1, 2.2, 2.3, and 2.4. The solid lines are the theoretical predictions, c_N∕t_sim, with c_N=(_⟨m⟩+−_⟨m⟩−)²var(x₁)=⟨2|m^|⟩2var(x₁) and $\text{var}\left(x_1\right)\equiv \text{var}\left(\overline{s}\right)$ defined in Eq. 8 for the MC data (N=1) and in Eq. 17 for the ST data (N>1). The dashed lines are fits of c_N∕t_sim. Note that for N=9, the actual error is slightly larger than predicted.