We study the convergence properties of Glauber dynamics for the random field Ising model (RFIM) with ferromagnetic interactions on finite domains of $\mathbb{Z}^d$, $d \ge 2$. Of particular interest is the Griffiths phase where correlations decay exponentially fast in expectation over the quenched disorder, but there exist arbitrarily large islands of weak fields where low-temperature behavior is observed. Our results are twofold: 1. Under weak spatial mixing (boundary-to-bulk exponential decay of correlations) in expectation, we show that the dynamics satisfy a weak Poincar\'e inequality implying algebraic relaxation to equilibrium over timescales polynomial in the volume $N$ of the domain, and polynomial time mixing from a warm start. From this we construct a polynomial-time approximate sampling algorithm based on running Glauber dynamics over an increasing sequence of approximations of the domain. 2. Under strong spatial mixing (exponential decay of correlations even near boundary pinnings) in expectation, we prove a full Poincar\'e inequality, implying exponential relaxation to equilibrium and $N^{o(1)}$-mixing time. Note by way of example, both weak and strong spatial mixing hold at any temperature, provided the external fields are strong enough. Our proofs combine a stochastic localization technique which has the effect of increasing the variance of the field, with a field-dependent coarse graining which controls the resulting sub-critical percolation process of sites with weak fields.

翻译：我们研究了$\mathbb{Z}^d$（$d \ge 2$）有限区域上具有铁磁相互作用的随机场伊辛模型（RFIM）的Glauber动力学收敛特性。特别关注Griffiths相，其中关联在淬火无序的期望下呈指数级快速衰减，但存在任意大的弱场区域，在这些区域中可观测到低温行为。我们的研究结果包含两个方面：1. 在期望意义下满足弱空间混合（边界到体关联的指数衰减）条件下，我们证明该动力学满足弱Poincaré不等式，这意味着在时间尺度上（与区域体积$N$呈多项式关系）存在代数弛豫至平衡态，以及从温和初始状态出发的多项式时间混合。基于此，我们通过在一系列逐步增大的区域近似上运行Glauber动力学，构建了一种多项式时间的近似采样算法。2. 在期望意义下满足强空间混合（即使在边界钉扎附近关联仍呈指数衰减）条件下，我们证明了完整的Poincaré不等式，这意味着指数级弛豫至平衡态以及$N^{o(1)}$的混合时间。举例而言，只要外场足够强，在任何温度下弱空间混合与强空间混合条件均成立。我们的证明结合了随机局域化技术（该技术具有增大场方差的效果）与场相关的粗粒化方法，后者可控制由此产生的弱场位点的亚临界渗流过程。