Some problems on mean-field spin glasses

自旋玻璃是理论物理学家S在70年代初发明的无序自旋系统，其目的是描述某些合金的奇怪磁性行为。在过去的几十年里，物理学家的研究产生了一系列的方法和预测。在数学界，自旋玻璃模型是纯粹的概率对象，呈现高度复杂的随机结构。这个研究项目的总体目标是对物理现象进行数学研究，包括随机哈密顿量的复杂性、吉布斯测度的结构和自由能的热力学极限。除了在概率领域很重要，这项研究计划中获得的工具和结果也与包括计算机科学、理论生物学和社会网络在内的各种科学分支有关。更准确地说，这项提议关注的是谢林顿-柯克帕特里克(Sherrington-Kirkpatrick，SK)模式。PI将研究函数序参数在低温区的结构和作用。此外，还将在Guera-TALAGRAND复本对称破缺界限下分析吉布斯度量的猜想超度性和温度中的混沌问题。PI还将利用Stein方法研究某些物理量的中心极限定理，如重叠和磁化强度，特别是在临界温度下。另一个方向是尾部严重紊乱的SK模式。由于现在允许更高级别的相互作用，预计该系统将出现不寻常的自旋玻璃现象。项目的最后部分涉及SK模型的两部分版本，该模型被用来模拟两个互动的大群体的集体属性，这些群体受到社会和神经网络、生物学和经济学的研究的推动。PI将集中在极限自由能的计算上。

Spin glasses are disordered spin systems invented by theoretical physicists in the early 70's. The aim is to describe the strange magnetic behaviors of certain alloys. Over the past decades, physicists' investigation has generated a series of methodologies and predictions. In the community of mathematics, spin glass models are purely probabilistic objects that present highly complex random structures. The general aim of this research project is to pursue mathematical investigation of physical phenomena including the complexity of the random Hamiltonian, the structure of the Gibbs measure and the thermodynamic limit of the free energy. As well as being important in the field of probability, the tools and results obtained in this research plan are relevant to a variety of scientific branches including computer science, theoretical biology and social networks. More precisely, this proposal is concerned about the Sherrington-Kirkpatrick (SK) model. The PI will investigate the structure and the role of the functional order parameter in the low temperature regime. In addition, the conjectured ultrametricity of the Gibbs measure and the problem of chaos in temperature will be analyzed following the Guerra-Talagrand replica symmetry breaking bound. The PI will as well study the central limit theorem for some physical quantities such as the overlap and magnetization especially at the critical temperature using Stein's method. Another direction is the SK model with heavy tail disorder. As now higher order interactions are allowed, the system has been predicted to exhibit unusual spin glass phenomena. The last part of the project involves the bipartite version of the SK model, which was used to model collective properties of two interactive large groups of individuals motivated by the studies of the social and neural networks, biology and economics. The PI will be concentrated on the computation of the limiting free energy.