## Bernoulli

• Bernoulli
• Volume 26, Number 3 (2020), 1863-1890.

### Logarithmic Sobolev inequalities for finite spin systems and applications

#### Abstract

We derive sufficient conditions for a probability measure on a finite product space (a spin system) to satisfy a (modified) logarithmic Sobolev inequality. We establish these conditions for various examples, such as the (vertex-weighted) exponential random graph model, the random coloring and the hard-core model with fugacity.

This leads to two separate branches of applications. The first branch is given by mixing time estimates of the Glauber dynamics. The proofs do not rely on coupling arguments, but instead use functional inequalities. As a byproduct, this also yields exponential decay of the relative entropy along the Glauber semigroup. Secondly, we investigate the concentration of measure phenomenon (particularly of higher order) for these spin systems. We show the effect of better concentration properties by centering not around the mean, but around a stochastic term in the exponential random graph model. From there, one can deduce a central limit theorem for the number of triangles from the CLT of the edge count. In the Erdős–Rényi model the first-order approximation leads to a quantification and a proof of a central limit theorem for subgraph counts.

#### Article information

Source
Bernoulli, Volume 26, Number 3 (2020), 1863-1890.

Dates
Revised: October 2019
First available in Project Euclid: 27 April 2020

Permanent link to this document
https://projecteuclid.org/euclid.bj/1587974526

Digital Object Identifier
doi:10.3150/19-BEJ1172

Mathematical Reviews number (MathSciNet)
MR4091094

Zentralblatt MATH identifier
07193945

#### Citation

Sambale, Holger; Sinulis, Arthur. Logarithmic Sobolev inequalities for finite spin systems and applications. Bernoulli 26 (2020), no. 3, 1863--1890. doi:10.3150/19-BEJ1172. https://projecteuclid.org/euclid.bj/1587974526

#### References

• [1] Adamczak, R. (2006). Moment inequalities for $U$-statistics. Ann. Probab. 34 2288–2314.
• [2] Adamczak, R., Kotowski, M., Polaczyk, B. and Strzelecki, M. (2019). A note on concentration for polynomials in the Ising model. Electron. J. Probab. 24 Paper No. 42, 22.
• [3] Adamczak, R. and Wolff, P. (2015). Concentration inequalities for non-Lipschitz functions with bounded derivatives of higher order. Probab. Theory Related Fields 162 531–586.
• [4] Bhamidi, S., Bresler, G. and Sly, A. (2011). Mixing time of exponential random graphs. Ann. Appl. Probab. 21 2146–2170.
• [5] Billingsley, P. (1968). Convergence of Probability Measures. New York: Wiley.
• [6] Bobkov, S.G. and Tetali, P. (2006). Modified logarithmic Sobolev inequalities in discrete settings. J. Theoret. Probab. 19 289–336.
• [7] Bonami, A. (1968). Ensembles $\Lambda (p)$ dans le dual de $D^{\infty }$. Ann. Inst. Fourier (Grenoble) 18 193–204.
• [8] Bonami, A. (1970). Étude des coefficients de Fourier des fonctions de $L^{p}(G)$. Ann. Inst. Fourier (Grenoble) 20 335–402.
• [9] Boucheron, S., Bousquet, O., Lugosi, G. and Massart, P. (2005). Moment inequalities for functions of independent random variables. Ann. Probab. 33 514–560.
• [10] Boucheron, S., Lugosi, G. and Massart, P. (2013). Concentration Inequalities: A nonasymptotic theory of independence. Oxford: Oxford Univ. Press.
• [11] Chatterjee, S. (2016). An introduction to large deviations for random graphs. Bull. Amer. Math. Soc. (N.S.) 53 617–642.
• [12] Chatterjee, S. and Diaconis, P. (2013). Estimating and understanding exponential random graph models. Ann. Statist. 41 2428–2461.
• [13] DeMuse, R., Easlick, T. and Yin, M. (2019). Mixing time of vertex-weighted exponential random graphs. J. Comput. Appl. Math. 362 443–459.
• [14] Diaconis, P. and Saloff-Coste, L. (1996). Logarithmic Sobolev inequalities for finite Markov chains. Ann. Appl. Probab. 6 695–750.
• [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] Erbar, M., Henderson, C., Menz, G. and Tetali, P. (2017). Ricci curvature bounds for weakly interacting Markov chains. Electron. J. Probab. 22 Paper No. 40, 23.
• [17] Gao, F. and Quastel, J. (2003). Exponential decay of entropy in the random transposition and Bernoulli–Laplace models. Ann. Appl. Probab. 13 1591–1600.
• [18] Götze, F., Sambale, H. and Sinulis, A. (2018). Concentration inequalities for bounded functionals via generalized log-Sobolev inequalities. Preprint. Available at arXiv:1812.01092.
• [19] Götze, F., Sambale, H. and Sinulis, A. (2019). Higher order concentration for functions of weakly dependent random variables. Electron. J. Probab. 24 Paper No. 85, 19.
• [20] Hanson, D.L. and Wright, F.T. (1971). A bound on tail probabilities for quadratic forms in independent random variables. Ann. Math. Stat. 42 1079–1083.
• [21] Jerrum, M. (1995). A very simple algorithm for estimating the number of $k$-colorings of a low-degree graph. Random Structures Algorithms 7 157–165.
• [22] Latała, R. (2006). Estimates of moments and tails of Gaussian chaoses. Ann. Probab. 34 2315–2331.
• [23] Ledoux, M. (2001). The Concentration of Measure Phenomenon. Mathematical Surveys and Monographs 89. Providence, RI: Amer. Math. Soc.
• [24] Marton, K. (2019). Logarithmic Sobolev inequalities in discrete product spaces. Combin. Probab. Comput. 28 919–935.
• [25] Mukherjee, S. (2013). Phase transition in the two star exponential random graph model. Preprint. Available at arXiv:1310.4164.
• [26] Mukherjee, S. (2013). Consistent estimation in the two star exponential random graph model. Preprint. Available at arXiv:1310.4526.
• [27] Nowicki, K. and Wierman, J.C. (1988). Subgraph counts in random graphs using incomplete $U$-statistics methods. In Proceedings of the First Japan Conference on Graph Theory and Applications (Hakone, 1986) 72 299–310.
• [28] Ruciński, A. (1988). When are small subgraphs of a random graph normally distributed? Probab. Theory Related Fields 78 1–10.
• [29] Rudelson, M. and Vershynin, R. (2013). Hanson–Wright inequality and sub-Gaussian concentration. Electron. Commun. Probab. 18 no. 82, 9.
• [30] Schudy, W. and Sviridenko, M. (2012). Concentration and moment inequalities for polynomials of independent random variables. In Proceedings of the Twenty-Third Annual ACM–SIAM Symposium on Discrete Algorithms 437–446. New York: ACM.
• [31] Talagrand, M. (1996). A new look at independence. Ann. Probab. 24 1–34.
• [32] van Handel, R. (2016). Probability in high dimension. APC 550 Lecture Notes, Princeton University.
• [33] Vigoda, E. (2001). A note on the Glauber dynamics for sampling independent sets. Electron. J. Combin. 8 Research Paper 8, 8.
• [34] Vu, V.H. (2002). Concentration of non-Lipschitz functions and applications. Random Structures Algorithms 20 262–316. Probabilistic methods in combinatorial optimization.
• [35] Wolff, P. (2013). On some Gaussian concentration inequality for non-Lipschitz functions. In High Dimensional Probability VI. Progress in Probability 66 103–110. Basel: Birkhäuser/Springer.
• [36] Wright, F.T. (1973). A bound on tail probabilities for quadratic forms in independent random variables whose distributions are not necessarily symmetric. Ann. Probab. 1 1068–1070.