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Curie-Weiss模型及其变体，例如具有随机相互作用。在这些环境中的著名结果包括平均自旋的弱收敛，称为磁化强度，以及高温下的高斯波动。然而，这是一个自然的问题，要求更精细的渐近性，包括大偏差的结果和收敛速度。我们喜欢应用累积量的方法，以及斯坦的方法，以获得精细的渐近这些模型中的磁化强度。对于经典的Curie-Weiss模型，通过Stein方法和模高斯收敛得到了Berry-Esseen界.混合矩的进一步渐近性是可用的。对于Curie-Weiss模型，其随机相互作用由Erdös-Rényi随机图建模，在淬火和退火设置下，也由Stein方法导出Berry-Esseen界限。对于块自旋伊辛模型中的块磁化矢量，高斯涨落之外的渐近性问题是开放的。然而，渐近矩是已知的。因此，累积量的方法，这已被证明是一个有用的工具，这种精细的渐近性，并在过去几年中获得的兴趣，可以导致进一步有趣的结果，并提供更深入的了解磁化的行为。由于伊辛模型中的长程相关性，累积量方法只在某些温度范围内产生Berry-Esseen边界。在我们的兴趣，更精细的渐近，我们相信，一个推广的斯坦的方法加权依赖图，一方面，允许收敛速度的失踪制度的伊辛模型，另一方面，扩展领域的应用斯坦的方法由大量的一类相依随机变量。

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.