For general ferromagnetic Ising models whose coupling matrix has bounded spectral radius, we show that the log-Sobolev constant satisfies a simple bound expressed only in terms of the susceptibility of the model. This bound implies very generally that the log-Sobolev constant is uniform in the system size up to the critical point (including on lattices), without using any mixing conditions. Moreover, if the susceptibility satisfies the mean-field bound as the critical point is approached, our bound implies that the log-Sobolev constant depends polynomially on the distance to the critical point and on the volume. In particular, this applies to the Ising model on subsets of $\mathbb{Z}^d$ when $d>4$. The proof uses a general criterion for the log-Sobolev inequality in terms of the Polchinski (renormalisation group) equation, a recently proved remarkable correlation inequality for Ising models with general external fields, the Perron--Frobenius theorem, and the log-Sobolev inequality for product Bernoulli measures.
翻译:对于总铁磁系模型,其组合矩阵将光谱半径捆绑在一起,我们显示,日志-Sobolev常量只满足一个简单的约束,仅以该模型的易感度表示。这一约束通常意味着日志-Sobolev常量在系统大小直到临界点(包括纬度)时是统一的,没有使用任何混合条件。此外,如果易感度满足了作为临界点被绑定的平均值,那么我们的约束意味着,日志-Sobolev常量在通往临界点和体积的距离上是多元的。特别是,这适用于当值为$\mathbb ⁇ d$的子集的Ising模型,而当值为$>4美元时。根据Polchinski(调整组)方程式,该证据使用了对日志-Sobolev不平等的一般标准,最近证明Ising模型与一般外部域、 Perron-Frobenius 理论,以及产品Bernoulli测量的log-Soblev不平等性。