Electronic Journal of Probability

Higher order concentration for functions of weakly dependent random variables

Friedrich Götze, Holger Sambale, and Arthur Sinulis

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Abstract

We extend recent higher order concentration results in the discrete setting to include functions of possibly dependent variables whose distribution (on the product space) satisfies a logarithmic Sobolev inequality with respect to a difference operator that arises from Glauber type dynamics. Examples include the Ising model on a graph with $n$ sites with general, but weak interactions (i.e. in the Dobrushin uniqueness regime), for which we prove concentration results of homogeneous polynomials, as well as random permutations, and slices of the hypercube with dynamics given by either the Bernoulli-Laplace or the symmetric simple exclusion processes.

Article information

Source
Electron. J. Probab., Volume 24 (2019), paper no. 85, 19 pp.

Dates
Received: 9 May 2018
Accepted: 25 June 2019
First available in Project Euclid: 10 September 2019

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1568080865

Digital Object Identifier
doi:10.1214/19-EJP338

Subjects
Primary: 60E15: Inequalities; stochastic orderings

Keywords
concentration of measure logarithmic Sobolev inequalities Ising model

Rights
Creative Commons Attribution 4.0 International License.

Citation

Götze, Friedrich; Sambale, Holger; Sinulis, Arthur. Higher order concentration for functions of weakly dependent random variables. Electron. J. Probab. 24 (2019), paper no. 85, 19 pp. doi:10.1214/19-EJP338. https://projecteuclid.org/euclid.ejp/1568080865


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References

  • [1] Radosław Adamczak, Moment inequalities for $U$-statistics, Ann. Probab. 34 (2006), no. 6, 2288–2314.
  • [2] Radosław Adamczak and Paweł Wolff, Concentration inequalities for non-Lipschitz functions with bounded derivatives of higher order, Probab. Theory Related Fields 162 (2015), no. 3–4, 531–586.
  • [3] Shigeki Aida and Daniel W. Stroock, Moment estimates derived from Poincaré and logarithmic Sobolev inequalities, Math. Res. Lett. 1 (1994), no. 1, 75–86.
  • [4] Luigi Ambrosio and Nicola Gigli, A user’s guide to optimal transport, Modelling and optimisation of flows on networks, Lecture Notes in Math., vol. 2062, Springer, Heidelberg, 2013, pp. 1–155.
  • [5] Luigi Ambrosio, Nicola Gigli, and Giuseppe Savaré, Gradient flows in metric spaces and in the space of probability measures, second ed., Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008.
  • [6] Sergey G. Bobkov, The growth of $L^{p}$-norms in presence of logarithmic Sobolev inequalities, Vestnik Syktyvkar Univ. 11 (2010), no. 2, 92–111.
  • [7] Sergey G. Bobkov, Friedrich Götze, and Holger Sambale, Higher order concentration of measure, Commun. Contemp. Math. 21 (2019), no. 3.
  • [8] Sergey G. Bobkov and Prasad Tetali, Modified logarithmic Sobolev inequalities in discrete settings, J. Theoret. Probab. 19 (2006), no. 2, 289–336.
  • [9] Guy Bresler and Dheeraj M. Nagaraj, Stein’s method for stationary distributions of Markov chains and application to Ising models, arXiv preprint (2017). arXiv:1712.05743
  • [10] Sourav Chatterjee and Partha S. Dey, Applications of Stein’s method for concentration inequalities, Ann. Probab. 38 (2010), no. 6, 2443–2485.
  • [11] Constantinos Daskalakis, Nishanth Dikkala, and Gautam Kamath, Concentration of multilinear functions of the Ising model with applications to network data, arXiv preprint (2017). arXiv:1710.04170
  • [12] Claude Dellacherie and Paul-André Meyer, Probabilities and potential, North-Holland Mathematics Studies, vol. 29, North-Holland Publishing Co., Amsterdam-New York; North-Holland Publishing Co., Amsterdam-New York, 1978.
  • [13] Amir Dembo and Ofer Zeitouni, Large deviations techniques and applications, Stochastic Modelling and Applied Probability, vol. 38, Springer-Verlag, Berlin, 2010, Corrected reprint of the second (1998) edition.
  • [14] Persi Diaconis and Laurent Saloff-Coste, Logarithmic Sobolev inequalities for finite Markov chains, Ann. Appl. Probab. 6 (1996), no. 3, 695–750.
  • [15] Reza Gheissari, Eyal Lubetzky, and Yuval Peres, Concentration inequalities for polynomials of contracting Ising models, Electron. Commun. Probab. 23 (2018), Paper No. 76, 12.
  • [16] Leonard Gross, Logarithmic Sobolev inequalities, Amer. J. Math. 97 (1975), no. 4, 1061–1083.
  • [17] Christof Külske, Concentration inequalities for functions of Gibbs fields with application to diffraction and random Gibbs measures, Comm. Math. Phys. 239 (2003), no. 1–2, 29–51.
  • [18] Rafał Latała, Estimates of moments and tails of Gaussian chaoses, Ann. Probab. 34 (2006), no. 6, 2315–2331.
  • [19] Tzong-Yow Lee and Horng-Tzer Yau, Logarithmic Sobolev inequality for some models of random walks, Ann. Probab. 26 (1998), no. 4, 1855–1873.
  • [20] Katalin Marton, Logarithmic Sobolev inequalities in discrete product spaces: a proof by a transportation cost distance, arXiv preprint (2015). arXiv:1507.02803
  • [21] Colin McDiarmid, On the method of bounded differences, Surveys in combinatorics, 1989 (Norwich, 1989), London Math. Soc. Lecture Note Ser., vol. 141, Cambridge Univ. Press, Cambridge, 1989, pp. 148–188.
  • [22] Elchanan Mossel, Ryan O’Donnell, and Krzysztof Oleszkiewicz, Noise stability of functions with low influences: invariance and optimality, Ann. of Math. (2) 171 (2010), no. 1, 295–341.
  • [23] Holger Sambale, Second order concentration for functions of independent random variables, Ph.D. thesis, Bielefeld University, 2016.
  • [24] Holger Sambale and Arthur Sinulis, Logarithmic Sobolev inequalities for finite spin systems and applications, arXiv preprint (2018). arXiv:1807.07765
  • [25] Daniel W. Stroock and Bogusław Zegarliński, The equivalence of the logarithmic Sobolev inequality and the Dobrushin–Shlosman mixing condition, Comm. Math. Phys. 144 (1992), no. 2, 303–323.
  • [26] Daniel W. Stroock and Bogusław Zegarliński, The logarithmic Sobolev inequality for discrete spin systems on a lattice, Comm. Math. Phys. 149 (1992), no. 1, 175–193.
  • [27] Cédric Villani, Optimal transport, old and new, Grundlehren der Mathematischen Wissenschaften, vol. 338, Springer-Verlag, Berlin, 2009.
  • [28] Bogusław Zegarliński, Dobrushin uniqueness theorem and logarithmic Sobolev inequalities, J. Funct. Anal. 105 (1992), no. 1, 77–111.