×

Quasi-factorization of the entropy and logarithmic Sobolev inequalities for Gibbs random fields. (English) Zbl 1086.82002

Summary: We show that the entropy functional exhibits a quasi-factorization property with respect to a pair of weakly dependent \(\sigma\)-algebras. As an application we give a simple proof that the Dobrushin and Shlosmans complete analyticity condition, for a Gibbs specification with finite range summable interaction, implies uniform logarithmic Sobolev inequalities. This result has been previously proven using several different techniques. The advantage of our approach is that it relies almost entirely on a general property of the entropy, while very little is assumed on the Dirichlet form. No topology is introduced on the single spin space, thus discrete and continuous spins can be treated in the same way.

MSC:

82B05 Classical equilibrium statistical mechanics (general)
39B62 Functional inequalities, including subadditivity, convexity, etc.
60G60 Random fields
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
PDFBibTeX XMLCite
Full Text: DOI