Logarithmic Sobolev inequalities for finite spin systems and applications

Sambale H, Sinulis A (2020)
BERNOULLI 26(3): 1863-1890.

Zeitschriftenaufsatz | E-Veröff. vor dem Druck | Englisch
 
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Abstract / Bemerkung
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 Erdos-Renyi model the first-order approximation leads to a quantification and a proof of a central limit theorem for subgraph counts.
Stichworte
central limit theorem; concentration of measure; exponential random; graph model; finite product spaces; logarithmic Sobolev inequality; mixing time; spin systems
Erscheinungsjahr
2020
Zeitschriftentitel
BERNOULLI
Band
26
Ausgabe
3
Seite(n)
1863-1890
ISSN
1350-7265
eISSN
1573-9759
Page URI
https://pub.uni-bielefeld.de/record/2943571

Zitieren

Sambale H, Sinulis A. Logarithmic Sobolev inequalities for finite spin systems and applications. BERNOULLI. 2020;26(3):1863-1890.
Sambale, H., & Sinulis, A. (2020). Logarithmic Sobolev inequalities for finite spin systems and applications. BERNOULLI, 26(3), 1863-1890. doi:10.3150/19-BEJ1172
Sambale, H., and Sinulis, A. (2020). Logarithmic Sobolev inequalities for finite spin systems and applications. BERNOULLI 26, 1863-1890.
Sambale, H., & Sinulis, A., 2020. Logarithmic Sobolev inequalities for finite spin systems and applications. BERNOULLI, 26(3), p 1863-1890.
H. Sambale and A. Sinulis, “Logarithmic Sobolev inequalities for finite spin systems and applications”, BERNOULLI, vol. 26, 2020, pp. 1863-1890.
Sambale, H., Sinulis, A.: Logarithmic Sobolev inequalities for finite spin systems and applications. BERNOULLI. 26, 1863-1890 (2020).
Sambale, Holger, and Sinulis, Arthur. “Logarithmic Sobolev inequalities for finite spin systems and applications”. BERNOULLI 26.3 (2020): 1863-1890.