无界长程势的局部中心极限定理
Local Central Limit Theorem for unbounded long-range potentials
AI总结：
本文证明了格点上长程相互作用两体势的积分中心极限定理与局部中心极限定理的等价性，利用集团展开和稀释技术分别处理无界与有界自旋情形。
摘要：
我们证明了在 $d\ge 1$ 的格点 $\mathbb{Z}^d$ 上具有长程相互作用的两体势的积分中心极限定理与局部中心极限定理之间的等价性。自旋空间可以是实轴上的任意子集，且可能是无界的，并配备一个合适的先验测度。对于一般的无界自旋，我们的方法在足够高的温度下有效，但对于有界自旋，我们的结果对每个温度都成立。我们的证明依赖于对积分特征函数的控制，这是通过将积分划分为三个不同区域来实现的，遵循四十年前由 Campanino、Del Grosso 和 Tirozzi 提出的标准方法。不同区域所需的界是通过集团展开技术获得的。对于有界自旋，温度的任意性是通过一种抽取（“稀释”）技术实现的，该技术同样在后一篇参考文献中引入。
英文摘要：
We prove the equivalence between the integral central limit theorem and the local central limit theorem for two-body potentials with long-range interactions on the lattice $\mathbb{Z}^d$ for $d\ge 1$. The spin space can be an arbitrary, possibly unbounded subset of the real axis with a suitable a-priori measure. For general unbounded spins, our method works at high-enough temperature, but for bounded spins our results hold for every temperature. Our proof relies on the control of the integrated characteristic function, which is achieved by dividing the integration into three different regions, following a standard approach proposed forty years ago by Campanino, Del Grosso and Tirozzi. The bounds required in the different regions are obtained through cluster-expansion techniques. For bounded spins, the arbitrariness of the temperature is achieved through a decimation ("dilution") technique, also introduced in the later reference.
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