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 -- equivalent to large-set expansion -- 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：我们研究具有铁磁相互作用的随机场伊辛模型（RFIM）在$\mathbb{Z}^d$（$d \ge 2$）有限区域上格劳伯动力学的收敛性质。特别关注Griffiths相，其中关联函数在淬火无序下的期望值呈指数衰减，但存在任意大的弱场区域，在这些区域内观察到低温行为。我们的结果有两个方面：1. 在期望值的弱空间混合（边界到内部的指数关联衰减）条件下，我们证明动力学满足弱Poincaré不等式——等价于大集扩张——这意味着在域体积$N$的多项式时间尺度上代数弛豫到平衡态，以及从暖启动出发的多项式时间混合。由此，我们构造了一个基于在域的递增近似序列上运行格劳伯动力学的多项式时间近似采样算法。2. 在期望值的强空间混合（即使在边界钉扎附近关联也呈指数衰减）条件下，我们证明完整的Poincaré不等式，这意味着指数弛豫到平衡态和$N^{o(1)}$混合时间。通过示例说明，只要外场足够强，弱空间混合和强空间混合都在任意温度下成立。我们的证明结合了随机局域化技术（其效果是增加场的方差）和场依赖粗粒化方法（控制由此产生的弱场位点的亚临界逾渗过程）。