We study the rate of convergence to equilibrium of the Glauber dynamics for the random-field Ising model on finite subsets of $\mathbb{Z}^d$ for $d\ge 2$. This is especially interesting in the so-called Griffiths phase, i.e., temperatures below the zero-field critical point where there is global decay of correlations induced by the random field, but there are local regions of small field values where correlations do not decay. We show under weak and strong assumptions on correlation decay in expectation, different Poincare-type inequalities and fast mixing guarantees. We will describe how the proofs combine the tools of stochastic localization schemes, with percolation-theoretic coarse-graining techniques. Based on joint work with A. El Alaoui, R. Eldan and A. Piana.
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