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2018 Concentration inequalities for polynomials of contracting Ising models
Reza Gheissari, Eyal Lubetzky, Yuval Peres
Electron. Commun. Probab. 23: 1-12 (2018). DOI: 10.1214/18-ECP173

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 $r \geq C \log N$.

Citation

Download Citation

Reza Gheissari. Eyal Lubetzky. Yuval Peres. "Concentration inequalities for polynomials of contracting Ising models." Electron. Commun. Probab. 23 1 - 12, 2018. https://doi.org/10.1214/18-ECP173

Information

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

zbMATH: 06964419
MathSciNet: MR3873783
Digital Object Identifier: 10.1214/18-ECP173

Subjects:
Primary: 60F05, 82B20, 82C20

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