Gaussian Fluctuations in Models of Statistical Mechanics – Fine Asymptotics for the Magnetization

在我们的项目中，我们研究了几种来自统计力学的模型，包括居里-魏斯模型及其变体，例如随机相互作用。在这些设置中，著名的结果包括平均自旋的弱收敛，称为磁化，以及高温下的高斯涨落。然而，要求更精细的渐近性是一个自然的问题，包括大偏差结果和收敛速度。我们喜欢采用累积量法和斯坦方法来获得这些模型中磁化强度的精细渐近。对于经典的Curie-Weiss模型，通过Stein方法和模态高斯收敛得到了Berry-Esseen界。混合矩的进一步渐近性是可用的。对于用Erdös-Rényi随机图建模的Curie-Weiss随机相互作用模型，在淬火和退火条件下均采用Stein方法推导出Berry-Esseen界。对于块自旋Ising模型中的块磁化矢量，高斯涨落以外的渐近性问题是开放的。然而，渐近矩是已知的。因此，累积量方法已被证明是这种精细渐近的有用工具，并在过去几年中引起了人们的兴趣，可以导致进一步有趣的结果，并提供对磁化行为的更深入的了解。由于Ising模型的长程依赖性，累积量的方法只在某些温度条件下产生Berry-Esseen边界。在我们对更精细的渐近性的兴趣中，我们确信将Stein的方法推广到加权依赖图，一方面，允许Ising模型中缺失区域的收敛率，另一方面，通过大量的相关随机变量扩展Stein的方法的应用领域。

In our project we investigate several models stemming from statistical mechanics including the Curie-Weiss model and variants thereof, e.g. with random interactions. Famous results in these settings include the weak convergence of the average spin, called the magnetization, and its Gaussian fluctuations for high temperatures. However, it is a natural question to ask for finer asymptotics including large deviation results and rates of convergence. We like to apply the method of cumulants as well as Stein's method in order to obtain fine asymptotics for the magnetization in these models. For the classical Curie-Weiss model Berry-Esseen bounds were derived by Stein's method and via mod-Gaussian convergence. Further asymptotics for mixed moments are available. For Curie-Weiss models with random interactions modeled by an Erdös-Rényi random graph also Berry-Esseen bounds were derived by Stein's method both in the quenched and annealed setting. For the vector of block magnetizations in the Block Spin Ising models, the question of asymptotics beyond Gaussian fluctuations are open. However, the asymptotic moments are known. Therefore the method of cumulants, which has proven to be a useful tool for such fine asymptotics and gained interest over the last years, can lead to further interesting results and provide deeper insight into the behavior of the magnetization. Due to the long-range dependence in the Ising model, the method of cumulants does yield Berry-Esseen bounds only in some temperature regimes. In our interest in finer asymptotics, we are convinced that a generalization of Stein's method to weighted dependency graphs will, on the one hand, allow convergence rates for the missing regimes in the Ising model and, on the other hand, extend the field of application of Stein's method by a substantial class of dependent random variables.