We consider the Ising model at its critical temperature with external magnetic field $ha^{15/8}$ on the square lattice with lattice spacing $a$. We show that the truncated two-point function in this model decays exponentially with a rate independent of $a$. As a consequence, we show exponential decay in the near-critical scaling limit Euclidean magnetization field. For the lattice model with $a=1$, the mass (inverse correlation length) is of order $h^{8/15}$ as $h\downarrow 0$; for the Euclidean field, it equals exactly $Ch^{8/15}$ for some $C$. Although there has been much progress in the study of critical scaling limits, results on near-critical models are far fewer due to the lack of conformal invariance away from the critical point. Our arguments combine lattice and continuum FK representations, including coupled conformal loop and measure ensembles, showing that such ensembles can be useful even in the study of near-critical scaling limits. Thus we provide the first substantial application of measure ensembles.

中文翻译：

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平面伊辛模型近临界尺度极限的指数衰减

我们考虑处于临界温度下的 Ising 模型，外部磁场 $ha^{15/8}$ 位于晶格间距 $a$ 的方形晶格上。我们表明，该模型中截断的两点函数以独立于 $a$ 的速率呈指数衰减。因此，我们在近临界尺度限制欧几里得磁化场中表现出指数衰减。对于$a=1$的晶格模型，质量（逆相关长度）的顺序为$h^{8/15}$为$h\downarrow 0$；对于欧几里得场，对于某些 $C$，它正好等于 $Ch^{8/15}$。尽管在临界尺度限制的研究方面取得了很大进展，但由于缺乏远离临界点的保形不变性，近临界模型的结果要少得多。我们的论点结合了点阵和连续 FK 表示，包括耦合共形环和测量集合，表明即使在研究近临界尺度限制时，这种集合也很有用。因此，我们提供了度量集合的第一个实质性应用。