文章基本信息

标题：Concentration inequalities for polynomials of contracting Ising models
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作者：Reza Gheissari ; Eyal Lubetzky ; Yuval Peres 等
期刊名称：Electronic Communications in Probability
印刷版ISSN：1083-589X
出版年度：2018
卷号：23
DOI：10.1214/18-ECP173
语种：English
出版社：Electronic Communications in Probability
摘要：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 $r \geq C \log N$.