论文标题
对近乎关键的ISING模型的log-sobolev不平等
Log-Sobolev inequality for near critical Ising models
论文作者
论文摘要
对于耦合矩阵的一般铁磁ising模型，我们表明log-sobolev常数满足仅根据模型的敏感性表示的简单结合。这种结合通常意味着系统大小的log-sobolev常数直至临界点（包括在晶格上），而无需使用任何混合条件。此外，如果易感性满足临界点接近临界点时的均值结合，我们的结合意味着log-sobolev常数在多个百膜仪上取决于到临界点和体积的距离。特别是，这适用于$ \ mathbb {z}^d $ $ d> 4 $的子集的ISING模型。 该证明使用Polchinski（肾小效应组）方程来使用一般标准，用于Log-Sobolev不平等，最近证明，对于具有一般外部领域，Perron-Frobenius Theorem和Log-Sobolev的ISING模型，最近被证明是显着的相关性不平等，以及产品Bernoulli Mesueres的log-Sobolev不平等。
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.
