Nonconvergence of the Wang-Landau algorithms with multiple random walkers

This paper discusses some convergence properties in the entropic sampling Monte Carlo methods with multiple random walkers, particularly in the Wang-Landau (WL) and $1/t$ algorithms. The classical algorithms are modified by the use of $m$-independent random walkers in the energy landscape to calculate the density of states (DOS). The Ising model is used to show the convergence properties in the calculation of the DOS, as well as the critical temperature, while the calculation of the number $\pi$ by multiple dimensional integration is used in the continuum approximation. In each case, the error is obtained separately for each walker at a fixed time, $t$; then, the average over $m$ walkers is performed. It is observed that the error goes as $1/\sqrt{m}$. However, if the number of walkers increases above a certain critical value $m>m_x$, the error reaches a constant value (i.e., it saturates). This occurs for both algorithms; however, it is shown that for a given system, the $1/t$ algorithm is more efficient and accurate than the similar version of the WL algorithm. It follows that it makes no sense to increase the number of walkers above a critical value $m_x$, since it does not reduce the error in the calculation. Therefore, the number of walkers does not guarantee convergence.

Figure 1

Behavior of the error as a function of time using the WL algorithm (long dashed line) and the $1/t$ algorithm (solid line) for a single walker. The data correspond to (a) the DOS and (b) the critical temperature, obtained from the peak location of the specific heat for a two-dimensional Ising model; and (c) the calculation of the number $\pi$ using multidimensional integrations. The critical time $t_x$, corresponding to the saturation of the error using the WL algorithm, is shown in the figures (vertical solid line). The times $t_1,t_2,t_3,t_4$, and $t_5$ correspond to the times at which the algorithm is stopped to start the $m$ walkers; these are indicated with vertical dashed lines. The slope of the curves corresponding to the $1/t$ algorithm goes as $1/\sqrt{t}$. The data represent the average of 200 independent realizations $(p=200)$.

Figure 2

The error as a function of the flatness criteria calculated for the WL algorithm. The curves are shown in decreasing order with 50%, 80%, 90%, 95%, 99%, and 99.9%, respectively. The data correspond to the critical temperature, obtained from the peak location of the specific heat for a two-dimensional Ising model with $L=8$, and it is the average of 200 independent realizations $(p=200)$.

Figure 3

(a) Behavior of $F\left(t\right)=lnf$, and (b) the error in the calculation of the critical temperature for a two-dimensional Ising model ${\epsilon }_{T_c}\left(t\right)$, calculated by using the WL algorithm for $80\%$ and $90\%$ flatness criteria, and $1/t$ algorithms. Vertical solid lines represent the saturation times for the WL algorithm $(t_x^{80}\approx 140000$ MCS and $t_x^{90}\approx 430000$ MCS). The times $t_A\approx 58000$ MCS, $t_B\approx 105000$ MCS, and $t_C\approx 3\times {10}^6$ MCS as $F\left(t\right)=ln{f}_{13}=1.2208\times {10}^{-4}$ are described in the text. The data represent the average of 200 independent realizations $(p=200)$.

Figure 4

Behavior of the mean value of the critical temperature as a function of time and the confidence interval for (a) the WL algorithm and (b) the $1/t$ algorithm (a magnification of the curve is shown in the inset). In (c), the behavior of the standard deviation for both algorithms is shown. The fitting of the $1/t$ curve gives a slope of $0.493\left(2\right)$. The data represent the average of 200 independent realizations $(p=200)$.

Figure 5

Best-fit Gaussians for the histograms of the critical temperatures obtained using (a) the WL algorithm with the 80%-flatness criterion, and (b) $1/t$ algorithms. Each of the curves corresponds to $p=100000$ independent runs. The curves are ordered from bottom to top according to the times defined in Fig. 1. Note that for the WL case, the curves collapse into each other for $t>t_x$.

Figure 6

Magnification of the curves described in Fig. 5. All the curves are normalized to 1 for comparison purposes.

Figure 7

Behavior of the error as a function of the number of walkers for the calculation of the DOS, using the WL algorithm (a) and the $1/t$ algorithm (b). In both figures, the curves are shown in decreasing order according to the times $t_1,t_2,t_x,t_3,t_4$, and $t_5$, which are defined in Fig. 1. The dotted line with slope $1/\sqrt{m}$ is shown for comparison. The data represent the average of 100 independent realizations $\left(p=100\right)$.

Figure 8

Behavior of the error as a function of the number of walkers for the calculation of the critical temperature using the WL algorithm (a) and the $1/t$ algorithm (b). The parameters are the same as in Fig. 7. The data represent the average of 100 independent realizations $\left(p=100\right)$.

Figure 9

Behavior of the error as a function of the number of the walkers for the calculation of number $\pi$, using the WL algorithm (Fig. 9) and the $1/t$ algorithm (Fig. 9). In both figures, the curves are shown in decreasing order according to the times $t_1,t_2,t_x,t_3,t_4$ and $t_5$, defined in Fig. 1. The parameters are the same as in Fig. 7. The data represent the average of 100 independent realizations $\left(p=100\right)$.

Figure 10

Behavior of the error as a function of the number of walkers for the calculation of the DOS for a window of 300 energy sites, corresponding to a two-dimensional Ising model with size $L=64$, using the WL algorithm (a) and the $1/t$ algorithm (b). The data represent the average of 100 independent realizations $\left(p=100\right)$.

Figure 11

(a) Behavior of the mean value of the critical temperature, $\overline{T_c}$, as a function of the number of walkers $m$ for the WL algorithm and the $1/t$ algorithm, calculated at fixed time $t=t_5$; the confidence interval is also shown for both curves. (b) Behavior of the standard deviation as a function of $m$ for both the WL and $1/t$ algorithms. Filled (empty) symbols represent $1/t$ (WL) procedures, respectively.

Figure 12

Density of state (DOS) for a fixed value of $t^{\prime }=10000$ MCS, and different values of $m$ $(m=10,20,30,40,\cdots ,200,2000$, from top to bottom in $E/L^2=0)$. In this case, the critical value of the number of walkers is $m_x=100$. The exact value of the DOS is denoted by a dotted line.