Electronic Communications in Probability

Concentration inequalities for polynomials of contracting Ising models

Abstract

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

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

Dates
Accepted: 1 October 2018
First available in Project Euclid: 19 October 2018

https://projecteuclid.org/euclid.ecp/1539914642

Digital Object Identifier
doi:10.1214/18-ECP173

Mathematical Reviews number (MathSciNet)
MR3873783

Zentralblatt MATH identifier
06964419

Citation

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

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