Critical Phenomena and Disorder Effects

该奖项将资助使用概率论数学工具的研究，以解决相变物理学中长期存在的问题，该领域被称为统计物理学。数学版本的概念的相变，阈值行为，临界现象，和标度限制的价值，这是现在公认的领域，乍一看可能似乎远离统计物理这些概念的起源。 对这些问题的数学研究导致了现代概率论的基本发展。 反过来，严格的分析为我们对相关物理学的理解提供了有用的反馈。 后者的例子是在二维和三维经典和量子系统系统中的相变的无序效应（如所谓的Imry-Ma现象），以及在随机量子算符背景下的无序的光谱和动力学效应中发现的。PI在过去的这类工作中发挥了重要作用，包括最近，他在其中贡献了关键结果，这些结果有助于发现新的经典和量子物理现象，这要归功于概率论的研究。 预计的研究将继续和重新定向PI的努力，并将继续必然是跨学科的性质。除了在物理问题上使用概率论和数学分析之外，研究结果可能会对现代概率论帮助阐明新现象的其他领域产生更广泛的影响，例如计算机科学和工程以及数据科学。 特别值得注意的是，该项目的其他更广泛的影响是在培训下一代美国科学家方面取得最高水平成果的潜力。基于PI过去和最近对NSF支持下的非常有才华的研究生和博士后的监督，该项目肯定会让PI为普林斯顿大学未来的学员提供最高水平的研究经验。一些研究将与其他机构的顶尖研究人员合作进行，为学员提供宝贵的联网机会。 PI的注意力将被重新引导到一些指导性模型中的临界现象上，这些临界现象低于它们的上临界尺寸，PI先前在渗流、伊辛自旋系统和φ ^[4]场论方面的工作坚定地推进了对这个概念的理解。 最近注意到，PI先前建立的伊辛模型在大于4维的尺度限制的高斯（玻色子）性质的技术也简单地证明了某些相关函数在平面情况下的费米性质。 这些关系中的一些已经知道，通过精确的解决方案的二维模型，但新的论点提出了一个路径，对“普遍”的紧急平面性在一类关键的非平面和不可解的二维模型的解释。 有关临界现象的相关问题，将探讨三维的情况下，这是既不平凡，也不解决，但明显的利益。 还将继续研究无序对随机算子的光谱和动力学性质的经典和量子效应，以及猝灭无序系统的吉布斯平衡态结构。

This award will fund research on the use of mathematical tools from probability theory to address long-standing questions in the physics of phase transitions, in an area known as statistical physics. Mathematical versions of the concepts of phase transitions, threshold behavior, critical phenomena, and scaling limits have value which is now well recognized in areas which at first sight might have seemed far from the statistical physics where these concepts originated. Mathematical studies of such topics have led to fundamental developments in modern probability theory. In turn, rigorous analysis has provided useful feedback on our understanding of the relevant physics. Examples of the latter are found in the disorder effects on phase transitions in two and three dimensional classical and quantum systems systems (such as the so-called Imry-Ma phenomenon), and in the spectral and dynamical effects of disorder in the context of random quantum operators. The PI was instrumental in past works of this type, including recently, in which he contributed key results which helped uncover new classical and quantum physical phenomena thanks to studies in probability theory. The projected research will both continue and redirect the PI's efforts, and will necessarily continue be interdisciplinary in nature. Beyond its use of probability theory and mathematical analysis on physics questions, the results of the research could potentially bring broader impacts to other areas where modern probability theory is helping elucidate new phenomena, such as computer science and engineering, and data science. Of particular note with this project's other broader impacts is the potential for outcomes of the highest caliber in training the next generation of US scientists. Based on the PI's past and recent supervision of extraordinarily talented graduate students and postdocs under NSF support, the project will certainly allow the PI to offer research experience of the highest level for such future trainees at Princeton University. Some of the research will be carried out in collaboration with top researchers from other institutions, providing valuable networking opportunities for the trainees. The PI's attention will be redirected towards critical phenomena in a number of instructive models below their upper critical dimension, a concept whose understanding was firmly advanced by PI's previous work on percolation, Ising spin systems and phi^4 field theory. It was recently noted that the techniques by which the PI has previously established the Gaussian (bosonic) nature of the Ising model's scaling limits in dimensions greater than four yield also simple proofs of the fermionic nature of certain correlation functions in the planar case. Some of these relations have been known through exact solution of the two dimensional model, but the new argument suggests a path towards explanation of "universal" emergent planarity in a class of critical non-planar and non-solvable two dimensional models. Related questions concerning critical phenomena will be explored for the three-dimensional case, which is neither trivial nor solvable yet of obvious interest. Work will also continue on classical and quantum effects of disorder on the spectral and dynamical properties of random operators, and on the structure of Gibbs equilibrium states of systems with quenched disorder.