Critical Phenomena and Disorder Effects

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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.