Electronic Communications in Probability

Concentration inequalities for polynomials of contracting Ising models

Reza Gheissari, Eyal Lubetzky, and Yuval Peres

Full-text: Open access


We study the concentration of a degree-$d$ polynomial of the $N$ spins of a general Ising model, in the regime where single-site Glauber dynamics is contracting. For $d=1$, Gaussian concentration was shown by Marton (1996) and Samson (2000) as a special case of concentration for convex Lipschitz functions, and extended to a variety of related settings by e.g., Chazottes et al. (2007) and Kontorovich and Ramanan (2008). For $d=2$, exponential concentration was shown by Marton (2003) on lattices. We treat a general fixed degree $d$ with $O(1)$ coefficients, and show that the polynomial has variance $O(N^d)$ and, after rescaling it by $N^{-d/2}$, its tail probabilities decay as $\exp (- c\,r^{2/d})$ for deviations of $r \geq C \log N$.

Article information

Electron. Commun. Probab., Volume 23 (2018), paper no. 76, 12 pp.

Received: 4 December 2017
Accepted: 1 October 2018
First available in Project Euclid: 19 October 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs 60F05: Central limit and other weak theorems 82C20: Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs

Ising model concentration of measure contraction independence testing

Creative Commons Attribution 4.0 International License.


Gheissari, Reza; Lubetzky, Eyal; Peres, Yuval. Concentration inequalities for polynomials of contracting Ising models. Electron. Commun. Probab. 23 (2018), paper no. 76, 12 pp. doi:10.1214/18-ECP173. https://projecteuclid.org/euclid.ecp/1539914642

Export citation


  • [1] J. R. Chazottes, P. Collet, C. Külske, and F. Redig. Concentration inequalities for random fields via coupling. Probability Theory and Related Fields, 137(1):201–225, 2007.
  • [2] M.-F. Chen. Trilogy of couplings and general formulas for lower bound of spectral gap. In Probability towards 2000 (New York, 1995), volume 128 of Lect. Notes Stat., pages 123–136. Springer, New York, 1998.
  • [3] C. Daskalakis, N. Dikkala, and G. Kamath. Testing Ising models. Extended abstract in Proc. of the 29th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2018), 1989–2007.
  • [4] H.-O. Georgii. Gibbs measures and phase transitions, volume 9 of De Gruyter Studies in Mathematics. Walter de Gruyter & Co., Berlin, second edition, 2011.
  • [5] G. Grimmett. The random-cluster model. In Probability on discrete structures, volume 110 of Encyclopaedia Math. Sci., pages 73–123. Springer, Berlin, 2004.
  • [6] L. A. Kontorovich and K. Ramanan. Concentration inequalities for dependent random variables via the martingale method. Ann. Probab., 36(6):2126–2158, 11 2008.
  • [7] C. Külske. Concentration inequalities for functions of gibbs fields with application to diffraction and random gibbs measures. Communications in Mathematical Physics, 239(1):29–51, 2003.
  • [8] M. Ledoux. The concentration of measure phenomenon, volume 89 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2001.
  • [9] D. A. Levin, Y. Peres, and E. L. Wilmer. Markov chains and mixing times. American Mathematical Society, Providence, RI, 2009.
  • [10] M. J. Luczak. Concentration of measure and mixing for Markov chains. In Fifth Colloquium on Mathematics and Computer Science, Discrete Math. Theor. Comput. Sci. Proc., AI, pages 95–120. Assoc. Discrete Math. Theor. Comput. Sci., Nancy, 2008.
  • [11] K. Marton. A measure concentration inequality for contracting Markov chains. Geom. Funct. Anal., 6(3):556–571, 1996.
  • [12] K. Marton. Measure concentration and strong mixing. Studia Sci. Math. Hungar., 40(1–2):95–113, 2003.
  • [13] E. J. McShane. Extension of range of functions. Bull. Amer. Math. Soc., 40(12):837–842, 12 1934.
  • [14] P.-M. Samson. Concentration of measure inequalities for Markov chains and $\Phi $-mixing processes. Ann. Probab., 28(1):416–461, 2000.
  • [15] H. Whitney. Analytic extensions of differentiable functions defined in closed sets. Transactions of the American Mathematical Society, 36(1):63–89, 1934.