Mean-Field Spin Glasses and Related Topics

在多个科学领域的数学建模中，展示大量平衡和崎岖景观的大型复杂系统是重要的。本项目将研究某些自旋玻璃模型。这些模型起源于某些合金的实验物理学，比如一种含有少量铁的金合金，具有不寻常的磁性行为。本项目旨在研究自旋玻璃模型的渐近结构和相变问题，这些问题是由高维优化、统计推断和神经网络等新兴主题所激发的。研究者将指导并与研究生和博士后合作进行这项研究。本工作的主要目的集中在某些平均场自旋玻璃模型及其稀释变体。这些问题包括理解与尖峰高斯矩阵模型和凹哈密顿量相关的自由能的吉布斯型变分公式。此外，还研究了Sherrington-Kirkpatrick模型和感知器模型在稀释情况下的高斯算子范数和极限自由能的渐近行为。本研究对高复杂度随机组合优化问题的建模具有重要的科学意义和应用价值。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Large complex systems exhibiting a huge number of equilibria and rugged landscapes are important in the mathematical modeling of multiple scientific areas. This project will study certain spin glass models. These models originated in the experimental physics of certain alloys, such as an alloy of gold with a small percentage of iron, possessing unusual magnetic behaviors. This project aims to investigate questions about asymptotic structures and phase transitions of spin glass models that are motivated by emerging topics in high-dimensional optimization, statistical inference, and neural networks. The investigator will mentor and collaborate with graduate students and postdoctoral fellows on this research.The main objective of this work focuses on certain mean-field spin glass models and their diluted variants. The problems include understanding the Gibbs-type variational formulation for the free energy associated with the spiked Gaussian matrix models and concave Hamiltonians. In addition, the asymptotic behavior of the Gaussian operator norm and the limiting free energy in the diluted setting of the Sherrington-Kirkpatrick model and perceptron models will be studied. This research is expected to be beneficial to the many scientific and applied subjects that can be modeled by high complexity randomized combinatorial optimization problems.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.