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量及其普适性是概率论长期追求的目标，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。对于相关性强的变量，问题就更复杂了。然而，在过去的二十年里，已经取得了重大的进展。就像概率学中的许多突破一样，新思想往往是从与物理学和统计力学的深度相互作用中涌现出来的。拟议的研究计划继续本着这一精神，旨在识别和描述高度相关变量的极值统计的普适性类别。重点放在高维和统计力学中出现的强相关性的高斯过程上。重要的例子包括自旋玻璃，如Sherrington-Kirkpatrick(SK)模型和Edwards-Anderson(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，物理学家提出了一系列大胆的猜想，以描述这类系统关联的复杂性。直到今天，这些问题在很大程度上仍然悬而未决。长期目标是严格确立和延伸这一理论的思想。今后五年的方向是：

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: