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Mean-value identities as an opportunity for Monte Carlo error reduction

Departamento de Física Teórica I, Universidad Complutense, 28040 Madrid, Spain
Instituto de Biocomputación y Física de Sistemas Complejos (BIFI), 50009 Zaragoza, Spain

Phys. Rev. E 79, 051109 – Published 11 May, 2009

DOI: https://doi.org/10.1103/PhysRevE.79.051109

Abstract

In the Monte Carlo simulation of both lattice field theories and of models of statistical mechanics, identities verified by exact mean values, such as Schwinger-Dyson equations, Guerra relations, Callen identities, etc., provide well-known and sensitive tests of thermalization bias as well as checks of pseudo-random-number generators. We point out that they can be further exploited as control variates to reduce statistical errors. The strategy is general, very simple, and almost costless in CPU time. The method is demonstrated in the two-dimensional Ising model at criticality, where the CPU gain factor lies between 2 and 4.

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References (30)

D. P. Landau and K. Binder, A Guide to Monte Carlo Simulations in Statistical Physics (Cambridge University Press, Cambridge, 2000).
A. D. Sokal, in Functional Integration: Basics and Applications (1996 Cargèse School), edited by C. DeWitt-Morette, P. Cartier, and A. Folacci (Plenum, New York, 1997).
G. Parisi, R. Petronzio, and F. Rapuano, Phys. Lett. B 128, 418 (1983).
M. Lüscher and P. Weisz, J. High Energy Phys. 09 (2001) 010.
See, e.g., F. Belletti et al. (the Janus Collaboration), Phys. Rev. Lett. 101, 157201 (2008).
J. M. Hammersley and D. C. Handscomb, Monte Carlo Methods (Fletcher and Son, Methuen, London, 1964).
R. Y. Rubinstein, Simulation and the Monte Carlo Method (Wiley and Sons, New York, 1981).
J. S. Schwinger, Proc. Natl. Acad. Sci. U.S.A. 37, 452 (1951); F. J. Dyson, Phys. Rev. 75, 1736 (1949).
H. B. Callen, Phys. Lett. 4, 161 (1963).
F. Guerra, Int. J. Mod. Phys. B 10, 1675 (1996).
R. H. Swendsen and J. S. Wang, Phys. Rev. Lett. 58, 86 (1987).
J. Salas and A. D. Sokal, J. Stat. Phys. 98, 551 (2000).
E. Marinari, G. Parisi, and J. J. Ruiz-Lorenzo, Phys. Rev. B 58, 14852 (1998).
H. G. Katzgraber, M. Palassini, and A. P. Young, Phys. Rev. B 63, 184422 (2001).
See, e.g., H. G. Ballesteros and V. Martin-Mayor, Phys. Rev. E 58, 6787 (1998), and references therein.
H. G. Ballesteros, L. A. Fernandez, V. Martin-Mayor, and A. Munoz Sudupe, Phys. Lett. B 387, 125 (1996); H. G. Ballesteros, L. A. Fernandez, V. Martin-Mayor, A. Munoz Sudupe, G. Parisi, and J. J. Ruiz-Lorenzo, Phys. Rev. B 58, 2740 (1998).
M. Weigel and W. Janke, Phys. Rev. Lett. 102, 100601 (2009).
For disordered systems, the time average in Eq. (9) is followed by a disorder average, which strongly diminishes the influence of the particular Monte Carlo dynamics.
D. Amit and V. Martin-Mayor, Field Theory: The Renormalization Group and Critical Phenomena, 3rd ed. (World-Scientific, Singapore, 2005).
S. S. Lavenberg and P. D. Welch, IBM Corporation, Yorktown Heights, New York Research Report No. RC8161, 1980 (unpublished).
A. E. Ferdinand and M. E. Fischer, Phys. Rev. 185, 832 (1969).
F. Cooper, B. Freedman, and D. Preston, Nucl. Phys. B 210, 210 (1982).
L. A. Fernandez, V. Martin-Mayor, and D. Yllanes, Nucl. Phys. B 807, 424 (2009).
L. A. Fernandez, V. Martin-Mayor, S. Perez-Gaviro, A. Tarancon, and A. P. Young e-print arXiv:0905.0322.
We shall need to recall the main features of the Swendsen-Wang dynamics, as formulated in Ref. [30]. The configuration space is expanded; to the original spins ${S_{\vec{x}}}$ in the lattice sites, we add bond variables $b_{\vec{x} \vec{y}} = 0, 1$ in the lattice bonds joining nearest-neighbor sites $\vec{x}$ and $\vec{y}$ (the marginal probability for the spins is the Boltzmann weight of the original Ising model). The $b = 1$ bonds are said to be occupied. Two lattice sites connected by a chain of occupied bonds are said to belong to the same cluster. The clusters partition the lattice into connected components. A crucial role is played by the conditional probabilities. That of the spins, given the bonds is: all the spins in a cluster take the same value, the spin values in different clusters are statistically independent and equal $\pm 1$ with 50% probability. For that of the bonds, given the spins, different bonds are statistically independent, and a given bond can be one only if it connects spins of equal sign and, in that case, with probability $1 - e^{- 2 κ}$ . The two Schrödinger-type operators, $P_{spin}$ and $P_{bond}$ belong to the heat-bath category; $P_{spin}$ $(P_{bond})$ leave the bonds (spins) unchanged and choose the new spin (bond) configuration according to the conditional probability of the spins (bonds), given the bonds (spins). As all heat-bath dynamics, both $P_{bond}$ and $P_{spin}$ verify detailed balance. Their product $P^{SW} = P_{spin} P_{bond}$ verifies only the softer balance condition, Eq. (10).
X. J. Li and A. D. Sokal, Phys. Rev. Lett. 63, 827 (1989).
Combining $⟨ η_{C}^{t} η_{S}^{t} ⟩ = 0$ with Eqs. (A3) and (A8),(A9),(A10), we recover the well-known cluster estimator for $U_{4}$ : $U_{4} = \frac{⟨ 3 {(\sum_{c} n_{c}^{2})}^{2} - 2 \sum_{c} n_{c}^{4} ⟩}{{⟨ \sum_{c} n_{c}^{2} ⟩}^{2}}$ .
For finite $L$ , $g_{C} > 0$ while the large- $L$ limit for $U_{4}$ is 3.
For instance, $⟨ M^{2} ⟩ = ⟨ P_{spin} M^{2} ⟩ = ⟨ C ⟩$ . A telegraphic proof of $C_{C B_{SW}} (0) = 0$ is also obtained by combining $⟨ C M^{2} ⟩ = ⟨ P_{spin} C M^{2} ⟩$ with $P_{spin} C M^{2} = C P_{spin} M^{2} = C^{2}$ and with Eq. (A18).
R. G. Edwards and A. D. Sokal, Phys. Rev. D 38, 2009 (1988).

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Mean-value identities as an opportunity for Monte Carlo error reduction

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