Abstract
We study the Glauber dynamics for the Ising model on the complete graph, also known as the Curie–Weiss Model. For β < 1, we prove that the dynamics exhibits a cut-off: the distance to stationarity drops from near 1 to near 0 in a window of order n centered at [2(1 − β)]−1 n log n. For β = 1, we prove that the mixing time is of order n 3/2. For β > 1, we study metastability. In particular, we show that the Glauber dynamics restricted to states of non-negative magnetization has mixing time O(n log n).
References
Aizenman, M., Holley, R.: Rapid convergence to equilibrium of stochastic Ising models in the Dobrushin Shlosman regime (Minneapolis, Minn., 1984–1985) IMA Vol. Math. Appl., vol. 8, pp. 1–11. Springer, New York (1987)
Aldous, D., Fill, J.: Reversible Markov chains and random walks on graphs (inprogress). Manuscript available at http://www.stat.berkeley.edu/~aldous/RWG/book.html
Bovier A., Eckhoff M., Gayrard V., Klein M.: Metastability in stochastic dynamics of disordered mean-field models. Probab. Theory Relat. Fields 119(1), 99–161 (2001)
Bovier A., Eckhoff M., Gayrard V., Klein M.: Metastability and low lying spectra in reversible Markov chains. Comm. Math. Phys. 228(2), 219–255 (2002)
Bovier A., Manzo F.: Metastability in Glauber dynamics in the low-temperature limit: beyond exponential asymptotics. J. Statist. Phys. 107(3–4), 757–779 (2002)
Bubley, R., Dyer, M.: Path coupling: a technique for proving rapid mixing in Markov chains. In: Proceedings of the 38th Annual Symposium on Foundations of Computer Science. I.E.E.E., Miami, FL, pp. 223–231 (1997)
Diaconis P.: The cutoff phenomenon in finite Markov chains. Proc. Natl. Acad. Sci. U.S.A. 93(4), 1659–1664 (1996)
Diaconis P., Saloff-Coste L.: Separation cut-offs for birth and death chains. Ann. Appl. Probab. 16(4), 2098–2122 (2006)
Ellis R.S.: Entropy, large deviations, and statistical mechanics. Grundlehren der Mathematischen Wissenschaften, vol. 271. Springer, New York (1985)
Ellis R.S., Newman C.M., Rosen J.S.: Limit theorems for sums of dependent random variables occurring in statistical mechanics. II. Conditioning, multiple phases, and metastability. Z. Wahrsch. Verw. Gebiete 51(2), 153–169 (1980)
Griffiths R.B., Weng C.-Y., Langer J.S.: Relaxation times for metastable states in the mean-field model of a ferromagnet. Phys. Rev. 149, 301–305 (1966)
Levin, D., Peres, Y., Wilmer, E.: Markov Chains and Mixing Times. American Mathematical Society, USA. Available at http://www.uoregon.edu/~dlevin/MARKOV/ (2008, to appear)
Lindvall, T.: Lectures on the coupling method. Dover Publications Inc., Mineola, NY. Corrected reprint of the 1992 original, pp. xiv+257 (2002)
Olivieri, E., Vares, M.E.: Large deviations and metastability, Encyclopedia of Mathematics and its Applications, vol. 100, pp. xvi+512. Cambridge University Press, Cambridge (2005)
Saloff-Coste, L.: Lectures on finite Markov chains, Lectures on Probability Theory and Statistics, Ecole d’Ete de Probabilites de Saint-Flour XXVI, 1996. Lecture Notes in Mathematics, vol. 1665, pp. 301–413. Springer, Berlin (1997)
Simon B., Griffiths R.B.: The \({(\varphi \sp{4})\sb{2}}\) field theory as a classical Ising model. Comm. Math. Phys. 33, 145–164 (1973)
Sinclair, A.: Algorithms for random generation and counting. In: Progress in Theoretical Computer Science. Birkhäuser Boston Inc., Boston, MA. A Markov chain approach, pp. vi+146 (1993)
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Research of Y. Peres was supported in part by NSF grant DMS-0605166.
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Levin, D.A., Luczak, M.J. & Peres, Y. Glauber dynamics for the mean-field Ising model: cut-off, critical power law, and metastability. Probab. Theory Relat. Fields 146, 223 (2010). https://doi.org/10.1007/s00440-008-0189-z
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DOI: https://doi.org/10.1007/s00440-008-0189-z
Keywords
- Markov chains
- Ising model
- Curie–Weiss model
- Mixing time
- Cut-off
- Coupling
- Glauber dynamics
- Metastability
- Heat-bath dynamics
- Mean-field model
Mathematics Subject Classification (2000)
- 60J10
- 60K35
- 82C20