Abstract
We develop transportation-entropy inequalities which are saturated by measures such that their log-density with respect to the background measure is an affine function, in the setting of the uniform measure on the discrete hypercube and the exponential measure. In this sense, this extends the well-known result of Talagrand in the Gaussian case. By duality, these transportation-entropy inequalities imply a strong integrability inequality for Bernoulli and exponential processes. As a result, we obtain on the discrete hypercube a dimension-free mean-field approximation of the free energy of a Gibbs measure and a nonlinear large deviation bound with only a logarithmic dependence on the dimension. Applied to the Ising model, we deduce that the mean-field approximation is within \(O(\sqrt{n} ||J||_2)\) of the free energy, where n is the number of spins and \(||J||_2\) is the Hilbert–Schmidt norm of the interaction matrix. Finally, we obtain a reverse log-Sobolev inequality on the discrete hypercube similar to the one proved recently in the Gaussian case by Eldan and Ledoux.
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Acknowledgements
I thank Ofer Zeitouni for many fruitful discussions which helped me build the present paper, as well as his valuable comments on an earlier version of the manuscript. I am grateful to Ronen Eldan for several influential and helpful discussions. Finally, I thank Michel Ledoux for his precious remarks which helped me improving this article. Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
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Augeri, F. A transportation approach to the mean-field approximation. Probab. Theory Relat. Fields 180, 1–32 (2021). https://doi.org/10.1007/s00440-021-01056-2
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DOI: https://doi.org/10.1007/s00440-021-01056-2