Critical Phenomena and Disorder Effects

该奖项将资助利用概率论中的数学工具来解决相变物理学中长期存在的问题的研究，该领域被称为统计物理学。相变、阈值行为、临界现象和尺度极限等概念的数学版本，在那些乍一看似乎与这些概念起源的统计物理学相距甚远的领域，现在已经得到了很好的认可。这些主题的数学研究导致了现代概率论的基本发展。反过来，严谨的分析为我们对相关物理学的理解提供了有用的反馈。后者的例子见于二维和三维经典系统和量子系统（如所谓的Imry-Ma现象）中对相变的无序效应，以及随机量子算符背景下无序的光谱和动力学效应。PI在过去的这类工作中发挥了重要作用，包括最近，他在概率论的研究中贡献了关键的结果，帮助揭示了新的经典和量子物理现象。计划中的研究将继续并改变PI的工作方向，并且必然继续具有跨学科的性质。除了在物理问题上使用概率论和数学分析之外，这项研究的结果可能会对现代概率论正在帮助阐明新现象的其他领域产生更广泛的影响，比如计算机科学与工程，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培养下一代美国科学家的最高水平的潜在结果。基于PI过去和最近在NSF支持下对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。部分研究将与其他机构的顶尖研究人员合作进行，为学员提供宝贵的交流机会。PI的注意力将被重新引导到一些低于其上临界维的指导性模型中的临界现象上，PI之前在渗透，伊辛自旋系统和phi^4场论方面的工作对这一概念的理解得到了有力的推进。最近有人注意到，PI先前所采用的技术已经在大于4维的情况下建立了Ising模型缩放极限的高斯（玻色子）性质，也产生了平面情况下某些相关函数的费米子性质的简单证明。其中一些关系已经通过二维模型的精确解得到，但新的论点提出了在一类临界非平面和不可解的二维模型中解释“普遍”涌现的平面性的途径。有关临界现象的相关问题将探讨三维情况，这既不是琐碎的，也不是可解决的，但明显的兴趣。我们还将继续研究无序对随机算符的光谱和动力学性质的经典和量子效应，以及具有淬灭无序的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.