Control of accuracy in the Wang-Landau algorithm

The Wang-Landau (WL) algorithm has been widely used for simulations in many areas of physics. Our analysis of the WL algorithm explains its properties and shows that the difference of the largest eigenvalue of the transition matrix in the energy space from unity can be used to control the accuracy of estimating the density of states. Analytic expressions for the matrix elements are given in the case of the one-dimensional Ising model. The proposed method is further confirmed by numerical results for the one-dimensional and two-dimensional Ising models and also the two-dimensional Potts model.

Figure 1

Dependence of $\overline{\delta }$ (solid line) and $\overline{\mathrm{\Delta }}$ (dotted line) on the Monte Carlo time $t$ for the $\text{WL}-1/t$ algorithm applied to the one-dimensional Ising model with $L=128$ (left panel) and to the two-dimensional Ising model on the square lattice of linear size $L=16$ (right panel) and with periodic boundary conditions. The vertical dashed line marks the average value of $t_s$.

Figure 2

Relative standard deviations $\sigma \left(\overline{\delta }\right)/\overline{\delta }$ (solid line) and $\sigma \left(\overline{\mathrm{\Delta }}\right)/\overline{\mathrm{\Delta }}$ (dotted line) as functions of $t$ for the simulations described in Fig. 1: $\sigma \left(\overline{\delta }\right)$ and $\sigma \left(\overline{\mathrm{\Delta }}\right)$ are standard deviations of the averaged values $\overline{\delta }$ and $\overline{\mathrm{\Delta }}$ obtained using 60 independent runs of the algorithm. Insets: $\sigma \left(\overline{\delta }\right)$ (solid line) and $\sigma \left(\overline{\mathrm{\Delta }}\right)$ (dotted line) as functions of $t$.

Figure 3

Dependence of $\overline{\delta }$ (solid line) and $\overline{\tilde{\mathrm{\Delta }}}$ (dotted line) on the Monte Carlo time $t$ for the $\text{WL}-1/t$ algorithm applied to the two-dimensional Potts model with $q=8$ spin states and with periodic boundary conditions. The lattice size is $L=32$ and $M=40$. Here, $\tilde{\mathrm{\Delta }}=1/N_E{\sum }_E|[\tilde{S}(E,t)-S_0\left(E\right)]/S_0\left(E\right)|$, where $S_0\left(E\right)=\langle S(E,t=2.6\times {10}^{12})\rangle$. The vertical dashed line marks the average value of $t_s$.

Figure 4

Dependence of $\overline{\delta }$ (solid line) and $\overline{\mathrm{\Delta }}$ (dotted line) on the Monte Carlo time $t$ for the $\text{WL}-1/t$ algorithm applied to the one-dimensional Ising model with $L=256$ (left panel) and $L=1024$ (right panel) and with periodic boundary conditions. The vertical dashed line marks the average value of $t_s$.

Figure 5

Dependence of $\overline{\delta }$ (solid line) and $\overline{\mathrm{\Delta }}$ (dotted line) on the Monte Carlo time $t$ for the $\text{WL}-1/t$ algorithm applied to the two-dimensional Ising model with $L=32$ (left panel) and $L=64$ (right panel) and with periodic boundary conditions. The vertical dashed line marks the average value of $t_s$.

Figure 6

Dependence of $\overline{\delta }$ (solid line) and $\overline{\mathrm{\Delta }}$ (dotted line) on the Monte Carlo time $t$ for the original WL algorithm applied to the one-dimensional Ising model with $L=32$ (left panel) and the two-dimensional Ising model with $L=32$ (right panel) and with periodic boundary conditions. We use $M=40$ in the left panel and $M=1$ in the right panel.