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arXiv · 2007.08200

A Spectral Condition for Spectral Gap: Fast Mixing in High-Temperature Ising Models

Ronen Eldan·Frederic Koehler·Ofer Zeitouni
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Abstract

We prove that Ising models on the hypercube with general quadratic interactions satisfy a Poincar\'{e} inequality with respect to the natural Dirichlet form corresponding to Glauber dynamics, as soon as the operator norm of the interaction matrix is smaller than $1$. The inequality implies a control on the mixing time of the Glauber dynamics. Our techniques rely on a localization procedure which establishes a structural result, stating that Ising measures may be decomposed into a mixture of measures with quadratic potentials of rank one, and provides a framework for proving concentration bounds for high temperature Ising models.

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Ronen Eldan, Frederic Koehler, Ofer Zeitouni. 2020-07-16. A Spectral Condition for Spectral Gap: Fast Mixing in High-Temperature Ising Models. https://arxiv.org/abs/2007.08200

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