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 when .

中文翻译：

---

近临界 Ising 模型的 Log-Sobolev 不等式

对于耦合矩阵具有有界谱半径的一般铁磁 Ising 模型，我们表明 log-Sobolev 常数满足仅以模型磁化率表示的简单界限。这个界限非常普遍地意味着，在不使用任何混合条件的情况下，log-Sobolev 常数在系统尺寸中直到临界点（包括在晶格上）都是均匀的。此外，如果磁化率在接近临界点时满足平均场界限，则我们的界限意味着 log-Sobolev 常数多项式取决于到临界点的距离和体积。特别是，这适用于子集上的 Ising 模型什么时候。