Abstract
We present rigorous results on the exponential convergence to equilibrium for the Swendsen-Wang stochastic dynamics for thed-dimensional Ising ferromagnet with external magnetic fieldh in the thermodynamic limit. We consider various situations, mainly in the low-temperature regime, in which boundary conditions are homogeneous and parallel or opposite to the external field. In the latter case we relate directly the tunneling from the metastable phase to the stable one with the exponential convergence to equilibrium.
This is a preview of subscription content, log in via an institution to check access.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
R. H. Swendsen and J. S. Wang, Nonuniversal critical dynamics in Monte Carlo simulations,Phys. Rev. Lett. 58 (1987).
Xiao-Jian-Li and A. D. Sokal, Rigorous lower bound on the dynamical critical exponent of the Swendsen-Wang dynamics,Phys. Rev. Lett. 63(8):827 (1989).
R. G. Edwards and A. D. Sokal, Generalization of the Fortuin-Kasteleyn-Swendsen-Wang representation and Monte Carlo algorithm,Phys. Rev. D 38(6):2009 (1988).
P. W. Kasteleyn and C. M. Fortuin, On the random cluster model. I. Introduction and relation to other models,Physica 57:536 (1972).
M. Aizenman, J. T. Chayes, L. Chayes, and C. M. Newman, The phase boundary in dilute and random Ising and Potts ferromagnets,J. Phys. A 20:L313 (1987).
R. Holley, Possible rates of convergence in finite range, attractive spin systems,Contemp. Math. 41 (1985).
R. Holley, Rapid convergence to equilibrium in one dimensional stochastic Ising models,Ann. Prob. 13:72–89 (1985).
F. Martinelli, E. Oliveri, and E. Scoppola, Metastability and exponential approach to equilibrium for low temperature stochastic Ising models, preprint, Rome (1989).
F. Martinelli, E. Oliveri, and E. Scoppola, On the Swendsen-Wang dynamics. II. Critical droplets and homogeneous nucleation at low temperature for the 2 dimensional Ising model,J. Stat. Phys. 62:135–159 (1991).
F. Martinelli and E. Scoppola, Small random perturbations of dynamical systems: Exponential loss of memory of the initial condition,Commun. Math. Phys. 120:25–69 (1988).
F. Martinelli, E. Oliveri, and E. Scoppola, Small random perturbations of finite and infinite dimensional dynamical system: Unpredictability of exit times,J. Stat. Phys. 55(3/4) (1989).
F. Martinelli, E. Oliveri, and E. Scoppola, On the loss of memory of initial conditions for some stochastic flows, inProceedings of the 2nd Ascona-Como International Conference, “Stochastic Process-Geometry & Physics,” S. Albeverio, ed. (World Scientific, Singapore).
R. Von Dreyfus, On the effect of randomness in ferromagnetic models and Schrödinger operators, Ph.D. Thesis, New York (1987).
A. Klein and R. Von Dreyfus, A new proof of localization in the Anderson tight binding model,Commun. Math. Phys. 124:285–299 (1989).
D. A. Huse and D. S. Fisher,Phys. Rev. B 35:6841 (1987).
A. D. Sokal and L. E. Thomas, Absence of mass gap for a class of stochastic contour models,J. Stat. Phys. 51:907 (1988).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Martinelli, F., Olivieri, E. & Scoppola, E. On the Swendsen-Wang dynamics. I. Exponential convergence to equilibrium. J Stat Phys 62, 117–133 (1991). https://doi.org/10.1007/BF01020862
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01020862