Statistics of Extrema in Disordered Systems and Related Models

自然界、物理学和社会中的许多现象都可以被看作是许多相似的成分或主体以看似随机或无序的方式相互作用的副产品（例如，股票市场、物理学中的凝聚态物质、数学中的素数）。这个项目的重点是研究在无序系统中出现的罕见事件或极端事件。在这些复杂的系统中存在着大量的因素和无序性，这使得直接的预测变得不可能。该项目的目标是开发概率论的工具，以提高我们对无序系统的基本理解，并最终导致对罕见事件的更好的统计建模，例如股票市场的高波动性事件和物理学中的某些类型的相变。具体来说，本项目的目的是将概率极值理论推广到具有强而复杂相关结构的随机系统。第一个目标是推导随机矩阵的特征多项式最大值和黎曼ζ函数（控制素数分布）的局部最大值的精细渐近性。这些系统是具有对数衰减相关性的随机过程的例子。第二部分是对自旋玻璃的极值的研究，自旋玻璃是一类重要的物理系统，具有更复杂的相关性，包括无序磁铁。主要目的是寻找一种新的方法来研究模型的吉布斯态结构。此外，将基于相关热力学量波动的扩展方法，研究有限维自旋玻璃（无序磁体的现实模型）相变的性质和存在性。

A wide range of phenomena in nature, physics, and society can be viewed as the byproduct of many similar components or agents interacting in a seemingly random or disordered manner (e.g., stock markets, condensed matter in physics, prime numbers in mathematics). The focus of this project is the study of rare events, or extrema, that emerge in disordered systems. The vast number of agents and the disorder present in these complex systems make straightforward predictions impossible. The project goal is to develop tools of probability theory to improve our fundamental understanding of disordered systems and, ultimately, to lead to better statistical modeling of rare events, such as high volatility episodes in stock markets and certain kinds of phase transitions in physics.Specifically, the aim of this project is to extend the theory of extreme values in probability to random systems with strong and complex correlation structure. The first objective is to derive fine asymptotics for the maxima of characteristic polynomials of random matrices and for the local maxima of the Riemann zeta function (which controls the distribution of prime numbers). These systems are examples of stochastic processes with logarithmically decaying correlations. The second part is a study of extrema of spin glasses, an important class of physical systems with more complex correlations that includes disordered magnets. The main objective there is to find a new approach to investigate the structure of the Gibbs states of the models. In addition, the nature and the existence of phase transitions for spin glasses in finite dimension (which are realistic models of disordered magnets) will be studied by extending methods based on fluctuations of relevant thermodynamic quantities.