Statistics of Extrema in Disordered Systems and Related Models

自然界、物理学和社会中的广泛现象可以被视为许多类似的成分或代理人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、数学中的质数)。这个项目的重点是研究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，存在着大量的代理人和无序，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中的某些类型的相变。具体地说，该项目的目的是将概率中的极值理论扩展到具有强而复杂的关联结构的随机系统。第一个目标是得到随机矩阵的特征多项式的极大值和Riemann Zeta函数(控制素数的分布)的局部极大值的精细渐近性。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极值的研究，这是一类重要的物理系统，具有更复杂的关联，其中包括无序磁铁。这里的主要目的是找到一种新的方法来研究模型的吉布斯态的结构。此外，还将利用基于相关热力学涨落的推广方法，研究有限维自旋玻璃(无序磁体的真实模型)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.