Annals of Probability

Applications of Stein’s method for concentration inequalities

Sourav Chatterjee and Partha S. Dey

Full-text: Open access


Stein’s method for concentration inequalities was introduced to prove concentration of measure in problems involving complex dependencies such as random permutations and Gibbs measures. In this paper, we provide some extensions of the theory and three applications: (1) We obtain a concentration inequality for the magnetization in the Curie–Weiss model at critical temperature (where it obeys a nonstandard normalization and super-Gaussian concentration). (2) We derive exact large deviation asymptotics for the number of triangles in the Erdős–Rényi random graph G(n, p) when p ≥ 0.31. Similar results are derived also for general subgraph counts. (3) We obtain some interesting concentration inequalities for the Ising model on lattices that hold at all temperatures.

Article information

Ann. Probab., Volume 38, Number 6 (2010), 2443-2485.

First available in Project Euclid: 24 September 2010

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60E15: Inequalities; stochastic orderings 60F10: Large deviations
Secondary: 60C05: Combinatorial probability 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)

Stein’s method Gibbs measures concentration inequality Ising model Curie–Weiss model large deviation Erdős–Rényi random graph exponential random graph


Chatterjee, Sourav; Dey, Partha S. Applications of Stein’s method for concentration inequalities. Ann. Probab. 38 (2010), no. 6, 2443--2485. doi:10.1214/10-AOP542.

Export citation


  • [1] Barthe, F., Cattiaux, P. and Roberto, C. (2005). Concentration for independent random variables with heavy tails. AMRX Appl. Math. Res. Express 2 39–60.
  • [2] Barthe, F., Cattiaux, P. and Roberto, C. (2006). Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry. Rev. Mat. Iberoamericana 22 993–1067.
  • [3] Bhamidi, S., Bresler, G. and Sly, A. (2008). Mixing time of exponential random graphs. In Proc. of the 49th Annual IEEE Symp. on FOCS 803–812. IEEE Computer Society, Washington, DC.
  • [4] Bobkov, S. G. (2007). Large deviations and isoperimetry over convex probability measures with heavy tails. Electron. J. Probab. 12 1072–1100 (electronic).
  • [5] Bobkov, S. G. and Ledoux, M. (2000). From Brunn–Minkowski to Brascamp–Lieb and to logarithmic Sobolev inequalities. Geom. Funct. Anal. 10 1028–1052.
  • [6] Bollobás, B. (2001). Random Graphs, 2nd ed. Cambridge Studies in Advanced Mathematics 73. Cambridge Univ. Press, Cambridge.
  • [7] Bolthausen, E. (1987). Laplace approximations for sums of independent random vectors. II. Degenerate maxima and manifolds of maxima. Probab. Theory Related Fields 76 167–206.
  • [8] Bolthausen, E., Comets, F. and Dembo, A. (2009). Large deviations for random matrices and random graphs. Unpublished manuscript.
  • [9] Boucheron, S., Lugosi, G. and Massart, P. (2003). Concentration inequalities using the entropy method. Ann. Probab. 31 1583–1614.
  • [10] Chatterjee, S. (2005). Concentration inequalities with exchangeable pairs. Ph.D. thesis, Stanford Univ. Available at arXiv:math/0507526.
  • [11] Chatterjee, S. (2007). Stein’s method for concentration inequalities. Probab. Theory Related Fields 138 305–321.
  • [12] Chatterjee, S. (2007). Concentration of Haar measures, with an application to random matrices. J. Funct. Anal. 245 379–389.
  • [13] Chatterjee, S. and Shao, Q.-M. (2009). Stein’s method of exchangeable pairs with application to the Curie–Weiss model. Preprint. Available at arXiv:0907.4450.
  • [14] Chazottes, J. R., Collet, P., Külske, C. and Redig, F. (2007). Concentration inequalities for random fields via coupling. Probab. Theory Related Fields 137 201–225.
  • [15] Döring, H. and Eichelsbacher, P. (2009). Moderate deviations in a random graph and for the spectrum of Bernoulli random matrices. Electron. J. Probab. 14 2636–2656.
  • [16] Eichelsbacher, P. and Lowe, M. (2009). Stein’s method for dependent random variables occurring in statistical mechanics. Preprint. Available at arXiv:0908.1909.
  • [17] Ellis, R. S. and Newman, C. M. (1978). The statistics of Curie–Weiss models. J. Stat. Phys. 19 149–161.
  • [18] Ellis, R. S. and Newman, C. M. (1978). Limit theorems for sums of dependent random variables occurring in statistical mechanics. Z. Wahrsch. Verw. Gebiete 44 117–139.
  • [19] Ellis, R. S. (1985). Entropy, Large Deviations, and Statistical Mechanics. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 271. Springer, New York.
  • [20] Gentil, I., Guillin, A. and Miclo, L. (2005). Modified logarithmic Sobolev inequalities and transportation inequalities. Probab. Theory Related Fields 133 409–436.
  • [21] Gozlan, N. (2007). Characterization of Talagrand’s like transportation-cost inequalities on the real line. J. Funct. Anal. 250 400–425.
  • [22] Gozlan, N. (2010). Poincare inequalities and dimension free concentration of measure. Ann. Inst. H. Poincaré Probab. Statist. To appear.
  • [23] Ising, E. (1925). Beitrag zur theorie des ferromagnetismus. Zeitschrift für Physik A Hadrons and Nuclei 31 253–258.
  • [24] Janson, S., Łuczak, T. and Rucinski, A. (2000). Random Graphs. Wiley, New York.
  • [25] Janson, S., Oleszkiewicz, K. and Ruciński, A. (2004). Upper tails for subgraph counts in random graphs. Israel J. Math. 142 61–92.
  • [26] Janson, S. and Ruciński, A. (2002). The infamous upper tail: Probabilistic methods in combinatorial optimization. Random Structures Algorithms 20 317–342.
  • [27] Kim, J. H. and Vu, V. H. (2004). Divide and conquer martingales and the number of triangles in a random graph. Random Structures Algorithms 24 166–174.
  • [28] Latała, R. and Oleszkiewicz, K. (2000). Between Sobolev and Poincaré. In Geometric Aspects of Functional Analysis. Lecture Notes in Math. 1745 147–168. Springer, Berlin.
  • [29] Ledoux, M. (2001). The Concentration of Measure Phenomenon. Mathematical Surveys and Monographs 89. Amer. Math. Soc., Providence, RI.
  • [30] Martin-Löf, A. (1982). A Laplace approximation for sums of independent random variables. Z. Wahrsch. Verw. Gebiete 59 101–115.
  • [31] Onsager, L. (1944). Crystal statistics. I. A two-dimensional model with an order–disorder transition. Phys. Rev. (2) 65 117–149.
  • [32] Park, J. and Newman, M. E. J. (2004). Statistical mechanics of networks. Phys. Rev. E (3) 70 066117–066122.
  • [33] Park, J. and Newman, M. E. J. (2005). Solution for the properties of a clustered network. Phys. Rev. E 72 026136–026137.
  • [34] Raič, M. (2007). CLT-related large deviation bounds based on Stein’s method. Adv. in Appl. Probab. 39 731–752.
  • [35] Simon, B. and Griffiths, R. B. (1973). The (ϕ4)2 field theory as a classical Ising model. Comm. Math. Phys. 33 145–164.
  • [36] Stein, C. (1972). A bound for the error in the normal approximation to the distribution of a sum of dependent random variables. In Proc. Sixth Berkeley Symp. Math. Statist. Probab. Vol. II: Probability Theory 583–602. Univ. California Press, Berkeley, CA.
  • [37] Stein, C. (1986). Approximate Computation of Expectations. IMS, Hayward, CA.
  • [38] Talagrand, M. (1995). Concentration of measure and isoperimetric inequalities in product spaces. Publ. Math. Inst. Hautes Études Sci. 81 73–205.
  • [39] Vu, V. H. (2001). A large deviation result on the number of small subgraphs of a random graph. Combin. Probab. Comput. 10 79–94.