Abstract
We study Glauber dynamics for the Ising model on the complete graph on n vertices, known as the Curie-Weiss Model. It is well known that at high temperature (β<1) the mixing time is Θ(nlog n), whereas at low temperature (β>1) it is exp (Θ(n)). Recently, Levin, Luczak and Peres considered a censored version of this dynamics, which is restricted to non-negative magnetization. They proved that for fixed β>1, the mixing-time of this model is Θ(nlog n), analogous to the high-temperature regime of the original dynamics. Furthermore, they showed cutoff for the original dynamics for fixed β<1. The question whether the censored dynamics also exhibits cutoff remained unsettled.
In a companion paper, we extended the results of Levin et al. into a complete characterization of the mixing-time for the Curie-Weiss model. Namely, we found a scaling window of order \(1/\sqrt{n}\) around the critical temperature β c =1, beyond which there is cutoff at high temperature. However, determining the behavior of the censored dynamics outside this critical window seemed significantly more challenging.
In this work we answer the above question in the affirmative, and establish the cutoff point and its window for the censored dynamics beyond the critical window, thus completing its analogy to the original dynamics at high temperature. Namely, if β=1+δ for some δ>0 with δ 2 n→∞, then the mixing-time has order (n/δ)log (δ 2 n). The cutoff constant is (1/2+[2(ζ2 β/δ−1)]−1), where ζ is the unique positive root of g(x)=tanh (β x)−x, and the cutoff window has order n/δ.
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References
Aizenman, M., Holley, R.: Rapid convergence to equilibrium of stochastic Ising models in the Dobrushin Shlosman regime. In: Percolation Theory and Ergodic Theory of Infinite Particle Systems, Minneapolis, Minn., 1984–1985. IMA Vol. Math. Appl., vol. 8, pp. 1–11. Springer, New York (1987)
Alon, N.: Eigenvalues and expanders. Combinatorica 6(2), 83–96 (1986). Theory of Computing (Singer Island, Fla., 1984)
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 (FOCS) pp. 223–231 (1997)
Chen, M.-F.: Trilogy of couplings and general formulas for lower bound of spectral gap. In: Probability Towards 2000, New York, 1995. Lecture Notes in Statist., vol. 128, pp. 123–136. Springer, New York (1998)
Ding, J., Lubetzky, E., Peres, Y.: The mixing time evolution of Glauber dynamics for the mean-field Ising model. Commun. Math. Phys. 289(2), 725–764 (2009)
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.: Limit theorems for sums of dependent random variables occurring in statistical mechanics. Z. Wahrsch. Verw. Geb. 44(2), 117–139 (1978)
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. Geb. 51(2), 153–169 (1980)
Freedman, D.A.: On tail probabilities for martingales. Ann. Probab. 3, 100–118 (1975)
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)
Jerrum, M., Sinclair, A.: Approximating the permanent. SIAM J. Comput. 18(6), 1149–1178 (1989)
Lawler, G.F., Sokal, A.D.: Bounds on the L 2 spectrum for Markov chains and Markov processes: a generalization of Cheeger’s inequality. Trans. Am. Math. Soc. 309(2), 557–580 (1988)
Levin, D.A., Peres, Y., Wilmer, E.: Markov Chains and Mixing Times. American Mathematical Society, Providence (2008)
Levin, D.A., Luczak, M., Peres, Y.: Glauber dynamics for the Mean-field Ising Model: cut-off, critical power law, and metastability. Probab. Theory Relat. Fields 146, 223–265 (2009)
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Ding, J., Lubetzky, E. & Peres, Y. Censored Glauber Dynamics for the Mean Field Ising Model. J Stat Phys 137, 407–458 (2009). https://doi.org/10.1007/s10955-009-9859-1
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DOI: https://doi.org/10.1007/s10955-009-9859-1