The Annals of Probability

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

Georg Menz and Felix Otto

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We consider a noninteracting 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, Villani, Westdickenberg and the second author 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 Cramér theorem.

Article information

Ann. Probab. Volume 41, Number 3B (2013), 2182-2224.

First available: 15 May 2013

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Mathematical Reviews number (MathSciNet)

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60J25: Continuous-time Markov processes on general state spaces 82B21: Continuum models (systems of particles, etc.)

Logarithmic Sobolev inequality spin system Kawasaki dynamics canonical ensemble coarse-graining


Menz, Georg; Otto, Felix. Uniform logarithmic Sobolev inequalities for conservative spin systems with super-quadratic single-site potential. The Annals of Probability 41 (2013), no. 3B, 2182--2224. doi:10.1214/11-AOP715.

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