Extreme Values of Highly Correlated Gaussian processes: a study of spin glasses and related models

理解随机过程中大值的统计及其普遍性是概率论的一个长期目标，在不同领域有着直接的应用。从20世纪30年代开始，极值理论已经确定了弱相依变量极值统计的普遍性质。对于具有强相关性的变量，问题更加复杂。然而，在过去20年中取得了重大进展。就像概率论中的许多突破一样，新的想法往往是从物理学和统计力学的深刻相互作用中产生的。拟议的研究计划继续在这种精神，旨在确定和描述的普适性类的高度相关的变量的极端值的统计。重点是在高维高斯过程和统计力学中出现的强相关性。重要的例子包括自旋玻璃，如谢灵顿-柯克帕特里克（SK）模型和爱德华-安德森（EA）模型，以及分支布朗运动（BBM）和高斯自由场（GFF）。20世纪80年代，物理学家们提出了一系列大胆的假设来描述这类系统的复杂性。这些问题直到今天仍然很大程度上没有解决。长期目标是严格建立和扩展理论的思想。未来五年的方向是： 1.为极值统计确定一个新的普适类，即对数相关模型类，它将包括BBM和GFF。 2.在吉布斯测度的水平上，找到变量之间接近最大值的相关性的通用结构，它集中在极值上。 3.对于Z^d上的爱德华-安德森（EA）模型，描述过程的吉布斯测度。特别是，建立d=2时模型的平凡性（无相变）和更高d时模型的非平凡性。 在这三个方向的进展将揭示核心原则，导致经典极值理论及其应用的扩展。

Understanding the statistics of large values in stochastic processes and their universality is a long-standing goal of probability theory with direct applications in diverse areas. Starting in the 1930s, extreme value theory has identified universal properties of the statistics of extreme values of weakly dependent variables. For variables with strong correlations, the problem is more complex. However, significant progress has been made in the past twenty years. Like many breakthroughs in probability, new ideas have often emerged from a deep interplay with physics and statistical mechanics. The proposed research program continues in this spirit, aiming to identify and describe universality classes for the statistics of extreme values of highly correlated variables. The focus is on Gaussian processes in high dimensions and with strong correlations that arise in statistical mechanics. Important examples include spin glasses, such as the Sherrington-Kirkpatrick (SK) model and the Edwards-Anderson (EA) model, as well as branching Brownian motion (BBM) and the Gaussian free field (GFF). A body of bold conjectures was proposed in the 1980s by physicists to describe the complexity of the correlations of such systems. These questions remain largely open to this day. The long-term objectives aim to rigorously establish and extend the ideas of the theory. The directions for the next five years are: 1. Identify a new universality class for extreme values statistics, the class of log-correlated models, that would include BBM and GFF. 2. Find universal structures for the correlations between the variables that are close to the maximum at the level of the Gibbs measure, which concentrates on the extreme values. 3. For the Edwards-Anderson (EA) model on Z^d, describe the Gibbs measure of the process. In particular, establish the triviality of the model at d=2 (no phase transition) and the non-triviality of the model for higher d. Progress in these three directions will unveil core principles leading to an extension of the classical extreme value theory and of its applications.