## Electronic Journal of Probability

### A variance inequality for Glauber dynamics applicable to high and low temperature regimes

Florian Völlering

#### Abstract

A variance inequality for spin-flip systems is obtained using comparatively weaker knowledge of relaxation to equilibrium based on coupling estimates for single site disturbances. We obtain variance inequalities interpolating between the Poincaré inequality and the uniform variance inequality, and a general weak Poincaré inequality. For monotone dynamics the variance inequality can be obtained from decay of the autocorrelation of the spin at the origin i.e. from that decay we conclude decay for general functions. This method is then applied to the low temperature Ising model, where the time-decay of the autocorrelation of the origin is extended to arbitrary quasi-local functions.

#### Article information

Source
Electron. J. Probab., Volume 19 (2014), paper no. 46, 21 pp.

Dates
Accepted: 31 May 2014
First available in Project Euclid: 4 June 2016

https://projecteuclid.org/euclid.ejp/1465065688

Digital Object Identifier
doi:10.1214/EJP.v19-2791

Mathematical Reviews number (MathSciNet)
MR3217334

Zentralblatt MATH identifier
1296.60265

Rights

#### Citation

Völlering, Florian. A variance inequality for Glauber dynamics applicable to high and low temperature regimes. Electron. J. Probab. 19 (2014), paper no. 46, 21 pp. doi:10.1214/EJP.v19-2791. https://projecteuclid.org/euclid.ejp/1465065688

#### References

• Aizenman, M.; Holley, R. Rapid convergence to equilibrium of stochastic Ising models in the Dobrushin Shlosman regime. Percolation theory and ergodic theory of infinite particle systems (Minneapolis, Minn., 1984-1985), 1–11, IMA Vol. Math. Appl., 8, Springer, New York, 1987.
• Chazottes, Jean-René; Redig, Frank; Völlering, Florian. The Poincaré inequality for Markov random fields proved via disagreement percolation. Indag. Math. (N.S.) 22 (2011), no. 3-4, 149–164.
• Dobrušin, R. L. Description of a random field by means of conditional probabilities and conditions for its regularity. (Russian) Teor. Verojatnost. i Primenen 13 1968 201–229.
• Holley, Richard. Possible rates of convergence in finite range, attractive spin systems. Particle systems, random media and large deviations (Brunswick, Maine, 1984), 215–234, Contemp. Math., 41, Amer. Math. Soc., Providence, RI, 1985.
• Holley, Richard; Stroock, Daniel. Logarithmic Sobolev inequalities and stochastic Ising models. J. Statist. Phys. 46 (1987), no. 5-6, 1159–1194.
• Kesten, Harry. Aspects of first passage percolation. École d'Été de probabilités de Saint-Flour, XIVâ€”1984, 125–264, Lecture Notes in Math., 1180, Springer, Berlin, 1986.
• Liggett, Thomas M. Interacting particle systems. Reprint of the 1985 original. Classics in Mathematics. Springer-Verlag, Berlin, 2005. xvi+496 pp. ISBN: 3-540-22617-6.
• Lubetzky, Eyal; Martinelli, Fabio; Sly, Allan; Toninelli, Fabio Lucio. Quasi-polynomial mixing of the 2D stochastic Ising model with "plus” boundary up to criticality. J. Eur. Math. Soc. (JEMS) 15 (2013), no. 2, 339–386.
• Martinelli, F.; Olivieri, E. Approach to equilibrium of Glauber dynamics in the one phase region. II. The general case. Comm. Math. Phys. 161 (1994), no. 3, 487–514.
• Röckner, Michael; Wang, Feng-Yu. Weak Poincaré inequalities and $L^ 2$-convergence rates of Markov semigroups. J. Funct. Anal. 185 (2001), no. 2, 564–603.
• Shlosman, S. B. Uniqueness and half-space nonuniqueness of Gibbs states in Czech models. (Russian) Teoret. Mat. Fiz. 66 (1986), no. 3, 430–444.
• van den Berg, J.; Maes, C. Disagreement percolation in the study of Markov fields. Ann. Probab. 22 (1994), no. 2, 749–763.
• Yoshida, Nobuo. Relaxed criteria of the Dobrushin-Shlosman mixing condition. J. Statist. Phys. 87 (1997), no. 1-2, 293–309.