论文标题
光谱间隙的光谱条件：高温伊辛模型中的快速混合
A Spectral Condition for Spectral Gap: Fast Mixing in High-Temperature Ising Models
论文作者
论文摘要
我们证明，一般性二次相互作用的超立方体上的Ising模型就可以在与Glauber Dynamics相对应的自然dirichlet形式上满足庞加莱的不平等，一旦交互矩阵的操作员规范小于$ 1 $。不平等意味着控制格劳伯动力学的混合时间。我们的技术依赖于建立结构结果的本地化程序，表明Ising措施可能被分解为具有二次势一的度量的混合物，并为证明高温ising模型的浓度界限提供了一个框架。
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.
