## Electronic Journal of Probability

### A note on concentration for polynomials in the Ising model

#### Abstract

We present precise multilevel exponential concentration inequalities for polynomials in Ising models satisfying the Dobrushin condition. The estimates have the same form as two-sided tail estimates for polynomials in Gaussian variables due to Latała. In particular, for quadratic forms we obtain a Hanson–Wright type inequality.

We also prove concentration results for convex functions and estimates for nonnegative definite quadratic forms, analogous as for quadratic forms in i.i.d. Rademacher variables, for more general random vectors satisfying the approximate tensorization property for entropy.

#### Article information

Source
Electron. J. Probab., Volume 24 (2019), paper no. 42, 22 pp.

Dates
Accepted: 17 February 2019
First available in Project Euclid: 17 April 2019

https://projecteuclid.org/euclid.ejp/1555466612

Digital Object Identifier
doi:10.1214/19-EJP280

Mathematical Reviews number (MathSciNet)
MR3949267

Zentralblatt MATH identifier
07055680

#### Citation

Adamczak, Radosław; Kotowski, Michał; Polaczyk, Bartłomiej; Strzelecki, Michał. A note on concentration for polynomials in the Ising model. Electron. J. Probab. 24 (2019), paper no. 42, 22 pp. doi:10.1214/19-EJP280. https://projecteuclid.org/euclid.ejp/1555466612

#### References

• [1] R. Adamczak, A note on the Hanson-Wright inequality for random vectors with dependencies, Electron. Commun. Probab. 20 (2015), no. 72, 13.
• [2] R. Adamczak, W. Bednorz, and P. Wolff, Moment estimates implied by modified log-Sobolev inequalities, ESAIM Probab. Stat. 21 (2017), 467–494.
• [3] R. Adamczak and R. Latała, Tail and moment estimates for chaoses generated by symmetric random variables with logarithmically concave tails, Ann. Inst. Henri Poincaré Probab. Stat. 48 (2012), no. 4, 1103–1136.
• [4] R. Adamczak and M. Strzelecki, On the convex Poincaré inequality and weak transportation inequalities, Bernoulli 25 (2019), no. 1, 341–374.
• [5] R. Adamczak and P. Wolff, Concentration inequalities for non-Lipschitz functions with bounded derivatives of higher order, Probab. Theory Related Fields 162 (2015), no. 3-4, 531–586.
• [6] S. Aida and D. Stroock, Moment estimates derived from Poincaré and logarithmic Sobolev inequalities, Math. Res. Lett. 1 (1994), no. 1, 75–86.
• [7] W. Beckner, Inequalities in Fourier analysis, Ann. of Math. (2) 102 (1975), no. 1, 159–182.
• [8] W. Bednorz and R. Latała, On the boundedness of Bernoulli processes, Ann. of Math. (2) 180 (2014), no. 3, 1167–1203.
• [9] S. G. Bobkov, The growth of ${L}_p$-norms in presence of logarithmic Sobolev inequalities, Vestnik Syktyvkar Univ. 11.2 (2010), 92–111.
• [10] A. Bonami, Étude des coefficients de Fourier des fonctions de $L^{p}(G)$, Ann. Inst. Fourier (Grenoble) 20 (1970), no. fasc. 2, 335–402 (1971).
• [11] P. Caputo, G. Menz, and P. Tetali, Approximate tensorization of entropy at high temperature, Ann. Fac. Sci. Toulouse Math. (6) 24 (2015), no. 4, 691–716.
• [12] S. Chatterjee, Stein’s method for concentration inequalities, Probab. Theory Related Fields 138 (2007), no. 1-2, 305–321.
• [13] J.-R. Chazottes, P. Collet, and F. Redig, On concentration inequalities and their applications for Gibbs measures in lattice systems, J. Stat. Phys. 169 (2017), no. 3, 504–546.
• [14] C. Daskalakis, N. Dikkala, and G. Kamath, Concentration of multilinear functions of the Ising model with applications to network data, Advances in Neural Information Processing Systems 30: Annual Conference on Neural Information Processing Systems 2017, 4–9 December 2017, Long Beach, CA, USA, 2017, pp. 12–22.
• [15] C. Daskalakis, N. Dikkala, and G. Kamath, Testing Ising models, Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, Philadelphia, PA, 2018, pp. 1989–2007.
• [16] L. Devroye, A. Mehrabian, and T. Reddad, The Minimax Learning Rate of Normal and Ising Undirected Graphical Models, ArXiv e-prints (2018).
• [17] P. Diaconis and L. Saloff-Coste, Logarithmic Sobolev inequalities for finite Markov chains, Ann. Appl. Probab. 6 (1996), no. 3, 695–750.
• [18] S. J. Dilworth and S. J. Montgomery-Smith, The distribution of vector-valued Rademacher series, Ann. Probab. 21 (1993), no. 4, 2046–2052.
• [19] R. Gheissari, E. Lubetzky, and Y. Peres, Concentration inequalities for polynomials of contracting Ising models, Electron. Commun. Probab. 23 (2018), Paper No. 76, 12 pp.
• [20] E. D. Gluskin and S. Kwapień, Tail and moment estimates for sums of independent random variables with logarithmically concave tails, Studia Math. 114 (1995), no. 3, 303–309.
• [21] F. Götze, H. Sambale, and A. Sinulis, Higher order concentration for functions of weakly dependent random variables, ArXiv e-prints (2018).
• [22] N. Gozlan, C. Roberto, P.-M. Samson, and P. Tetali, Kantorovich duality for general transport costs and applications, J. Funct. Anal. 273 (2017), no. 11, 3327–3405.
• [23] L. Gross, Logarithmic Sobolev inequalities, Amer. J. Math. 97 (1975), no. 4, 1061–1083.
• [24] D. L. Hanson and F. T. Wright, A bound on tail probabilities for quadratic forms in independent random variables, Ann. Math. Statist. 42 (1971), 1079–1083.
• [25] P. Hitczenko, S. Kwapień, On the Rademacher series. Probability in Banach spaces, 9 (Sandjberg, 1993), 31–36, Progr. Probab., 35, Birkhäuser Boston, Boston, MA, 1994.
• [26] E. Ising, Beitrag zur Theorie des Ferromagnetismus, Zeitschrift fur Physik 31 (1925), 253–258.
• [27] R. Latała, Tail and moment estimates for some types of chaos, Studia Math. 135 (1999), no. 1, 39–53.
• [28] R. Latała, Estimates of moments and tails of Gaussian chaoses, Ann. Probab. 34 (2006), no. 6, 2315–2331.
• [29] M. Ledoux, The concentration of measure phenomenon, Mathematical Surveys and Monographs, vol. 89, American Mathematical Society, Providence, RI, 2001.
• [30] M. J. Łuczak, Concentration of measure and mixing for Markov chains, Fifth Colloquium on Mathematics and Computer Science, Discrete Math. Theor. Comput. Sci. Proc., AI, Assoc. Discrete Math. Theor. Comput. Sci., Nancy, 2008, pp. 95–120.
• [31] K. Marton, Measure concentration and strong mixing, Studia Sci. Math. Hungar. 40 (2003), no. 1-2, 95–113.
• [32] K. Marton, Logarithmic Sobolev inequalities in discrete product spaces: a proof by a transportation cost distance, ArXiv e-prints (2015).
• [33] S. J. Montgomery-Smith, The distribution of Rademacher sums, Proc. Amer. Math. Soc. 109 (1990), no. 2, 517–522.
• [34] E. Nelson, Construction of quantum fields from Markoff fields, J. Functional Analysis 12 (1973), 97–112.
• [35] R. O’Donnell, Analysis of Boolean functions, Cambridge University Press, New York, 2014.
• [36] P.M. Samson, Concentration of measure inequalities for Markov chains and $\Phi$-mixing processes, Ann. Probab. 28 (2000), no. 1, 416–461.
• [37] N. P. Santhanam and M. J. Wainwright, Information-theoretic limits of selecting binary graphical models in high dimensions, IEEE Trans. Inform. Theory 58 (2012), no. 7, 4117–4134.
• [38] K. Shanmugam, R. Tandon, A. G. Dimakis, and P. Ravikumar, On the Information Theoretic Limits of Learning Ising Models, Proceedings of the 27th International Conference on Neural Information Processing Systems, volume 2 of NIPS’14. Cambridge, MA, USA, 2014. MIT Press (2014), 2303–2311.
• [39] Y. Shu and M. Strzelecki, A characterization of a class of convex log-Sobolev inequalities on the real line, Ann. Inst. Henri Poincaré Probab. Stat. 54 (2018), no. 4, 2075–2091.
• [40] M. Talagrand, Concentration of measure and isoperimetric inequalities in product spaces, Inst. Hautes Études Sci. Publ. Math. (1995), no. 81, 73–205.
• [41] M. Talagrand, New concentration inequalities in product spaces, Invent. Math. 126 (1996), no. 3, 505–563.
• [42] M. Talagrand, Upper and lower bounds for stochastic processes, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 60, Springer, Heidelberg, 2014, Modern methods and classical problems.
• [43] R. Vershynin, High-dimensional probability: An introduction with applications in data science, Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press, 2018.