Gaussian Fluctuations in Models of Statistical Mechanics – Fine Asymptotics for the Magnetization
Description Phase 2
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 Erdős-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 behaviour 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.