References

1

For a review, see: ``The Monte Carlo Method in Condensed Matter Physics,'' ed. by K. Binder (Springer, Berlin, Heidelberg 1992).

2

H. Müller-Krumbhaar and K. Binder, J. Stat. Phys. 8, 1 (1973).

3

N. Madras and A. D. Sokal, J. Stat. Phys. 50, 109 (1988).

4

A.M. Ferrenberg, D.P. Landau and K. Binder, J. Stat. Phys 63, 867 (1991).

5

A.M. Ferrenberg and R.H. Swendsen, Phys. Rev. Lett. 61, 2635 (1988); A.M. Ferrenberg and R.H. Swendsen, Phys. Rev. Lett. 63, 1195 (1989).

6

R.H. Swendsen, J.-S. Wang and A.M. Ferrenberg, ``New Monte Carlo Methods for Improved Efficiency of Computer Simulations'' in reference [1].

7

A.M. Ferrenberg and D.P. Landau, Phys. Rev. B 44, 5081 (1991).

8

An old but excellent reference is: W. Feller, ``An Introduction to Probability Theory and Its Applications,'' (Wiley, New York 1957).

9

The definition of the integrated correlation time used here differs from that used in some recent papers. See, for example reference [3].

10

See for example: P.R. Bevington, ``Data reduction and error analysis for the physical sciences,'' (McGraw-Hill, New York 1969).

11

E. Ising, Z. Phys. 31, 2553 (1925).

12

L. Onsager, Phys. Rev 65, 117 (1944).

13

A.E. Ferdinand and M.E. Fisher, Phys. Rev 185, 832 (1969).

14

N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller, and E. Teller, J. Chem. Phys. 21, 1087 (1953).

15

U. Wolff, Phys. Rev. Lett. 62, 361 (1989).

16

C.-K. Hu, Phys. Rev. Lett. 69, 2739 (1992).

17

P.W. Kasteleyn and C.M. Fortuin, J. Phys. Soc. Jpn. Suppl. 26s, 11 (1969); C.M. Fortuin and P.W. Kasteleyn, Physicsa (Utrecht) 57, 536 (1972).

18

W. Janke, unpublished.

19

E.Münger and M. Novotny, Phys. Rev. B 43, 5773 (1991).

20

B.Berg and T. Neuhaus, Phys. Rev. Lett. 68, 9 (1992).

21

B.Berg and T. Celik, Phys. Rev. Lett. 69, 2292 (1992).

22

W. Janke and T. Sauer, Phys. Rev. E 49, 3475 (1994).

23

A.M. Ferrenberg and D.P. Landau, unpublished.

Figure Captions

Fig. 1: Absolute error in E as a function of for the d=2 Ising model simulated at infinite temperature (). The measured error is compared to the theoretical error from (15) as well as the Gaussian test model (16).

Fig. 2: Relative error in C as a function of for the d=2 Ising model simulated at infinite temperature (). The measured error is compared to the theoretical error from (15) as well as the Gaussian test model (17).

Fig. 3: Comparison of the theoretical and measured relative error in E as a function of for the d=2 Ising model simulated at (marked by the vertical line). The simulations were performed with the Metropolis algorithm.

Fig. 4: Comparison of the theoretical and measured relative error in C as a function of for the d=2 Ising model simulated at . The vertical line indicates the simulated temperature. Results obtained using the Metropolis algorithm are shown.

Fig. 5: Ratio of the deviation from the correct answer to the statistical error for the d=2 Ising model simulated at (marked by the vertical line) using the Metropolis algorithm. The horizontal lines represent standard deviation.

Fig. 6: Comparison of the relative error in the energy for a simulation with independent measurements to a Metropolis simulation with correlations. The vertical line indicates the location of the simulated temperature. Results for the d=2 Ising model are shown.

Fig. 7: Comparison of the relative error in E determined by Metropolis and Wolff simulations for the d=2 Ising model. To simplifiy the comparison, the error from the Wolff algorithm was rescaled to match that of the Metropolis algorithm at the simulated temperature (marked by the vertical line).

Fig. 8: Plot of the reweighted energy correlation time () for the Metropolis and Wolff algorithms. Results for the d=2 Ising model are shown. The simulations were performed at , which is marked by the vertical line.