Uniform logarithmic Sobolev inequalities for conservative spin systems with super-quadratic single-site potential.

Georg Menz and Felix Otto

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Submission date: 07. Feb. 2011
Pages: 48
published in: The annals of probability, 41 (2013) 3B, p. 2182-2224 
DOI number (of the published article): 10.1214/11-AOP715
Bibtex
MSC-Numbers: 60K35, 60J25, 82B21
Keywords and phrases: Logarithmic Sobolev inequality, Spin system, Kawasaki dynamics, Canonical ensemble, coarse-graining
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Abstract:
We consider a non-interacting unbounded spin system with conservation of the mean spin. We derive a uniform logarithmic Sobolev inequality (LSI) provided the single-site potential is a bounded perturbation of a strictly convex function. The scaling of the LSI constant is optimal in the system size. The argument adapts the two-scale approach of Grunewald, Otto, Westdickenberg, and Villani from the quadratic to the general case. Using an asymmetric Brascamp-Lieb type inequality for covariances we reduce the task of deriving a uniform LSI to the convexification of the coarse-grained Hamiltonian, which follows from a general local Cramèr theorem.