We derive entropy factorization estimates for spin systems using the stochastic localization approach proposed by Eldan and Chen-Eldan, which, in this context, is equivalent to the renormalization group approach developed independently by Bauerschmidt, Bodineau, and Dagallier. The method provides approximate Shearer-type inequalities for the corresponding Gibbs measure at sufficiently high temperature, without restrictions on the degree of the underlying graph. For Ising systems, these are shown to hold up to the critical tree-uniqueness threshold, including polynomial bounds at the critical point, with optimal $O(\sqrt n)$ constants for the Curie-Weiss model at criticality. In turn, these estimates imply tight mixing time bounds for arbitrary block dynamics or Gibbs samplers, improving over existing results. Moreover, we establish new tensorization statements for the Shearer inequality asserting that if a system consists of weakly interacting but otherwise arbitrary components, each of which satisfies an approximate Shearer inequality, then the whole system also satisfies such an estimate.

翻译：我们利用Eldan以及Chen-Eldan提出的随机局部化方法，推导了自旋系统的熵因子化估计。在此背景下，该方法等价于Bauerschmidt、Bodineau和Dagallier独立发展的重整化群方法。该方法为足够高温下对应的吉布斯测度提供了近似的Shearer型不等式，且不受底层图度数的限制。对于伊辛系统，这些不等式被证明在临界树唯一性阈值以下均成立，包括临界点的多项式界，且在临界Curie-Weiss模型中具有最优的$O(\sqrt n)$常数。反过来，这些估计意味着任意块动力学或吉布斯采样器的紧致混合时间界，改进了现有结果。此外，我们为Shearer不等式建立了新的张量化陈述：如果一个系统由弱相互作用但其他方面任意的组件构成，且每个组件都满足一个近似的Shearer不等式，则整个系统也满足这样的估计。