A new criterion for the logarithmic Sobolev inequality and two applications

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Abstract

We give a criterion for the logarithmic Sobolev inequality (LSI) on the product space X1 × ⋯ × XN. We have in mind an N-site lattice, unbounded continuous spin variables, and Glauber dynamics. The interactions are described by the Hamiltonian H of the Gibbs measure. The criterion for LSI is formulated in terms of the LSI constants of the single-site conditional measures and the size of the off-diagonal entries of the Hessian of H. It is optimal for Gaussians with positive covariance matrix. To illustrate, we give two applications: one with weak interactions and one with strong interactions and a decay of correlations condition. © 2006 Elsevier Inc. All rights reserved.

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Otto, F., & Reznikoff, M. G. (2007). A new criterion for the logarithmic Sobolev inequality and two applications. Journal of Functional Analysis, 243(1), 121–157. https://doi.org/10.1016/j.jfa.2006.10.002

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