Probability Theory and Related Fields ( IF 2 ) Pub Date : 2021-08-20 , DOI: 10.1007/s00440-021-01085-x Ronen Eldan 1 , Ofer Zeitouni 1 , Frederic Koehler 2
We prove that Ising models on the hypercube with general quadratic interactions satisfy a Poincaré 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.
中文翻译:
光谱间隙的光谱条件:高温 Ising 模型中的快速混合
我们证明,只要相互作用矩阵的算子范数小于 1,具有一般二次相互作用的超立方体上的 Ising 模型就满足 Poincaré 不等式关于对应于 Glauber 动力学的自然 Dirichlet 形式。该不等式意味着对芒硝动力学的混合时间。我们的技术依赖于建立结构结果的定位程序,说明 Ising 度量可以分解为具有 1 阶二次势的度量的混合,并提供了一个框架来证明高温 Ising 模型的浓度界限。