The PI will investigate a variety of projects on the borderline of probability theory and statistical mechanics. A common feature of these is their focus on overcoming the technical obstacles that have insofar held progress to the mathematical understanding of the underlying physical phenomena. The first set of specific problems studies the role of spectral characteristics of random lattice Laplacians in the derivation of scaling limits of random walks in disordered media. The second project investigates the effects of introducing non-attractive potentials on the statics and dynamics of random gradient fields. The third project focuses on developing a detailed approach to localization of random walks in a random potential landscape based on eigenvalue extreme order statistics for lattice Anderson Hamiltonians. The fourth project utilizes new ideas to control the rigidity of interfaces in statistical mechanical models. The final project outlines a new approach based on exchangeability to quantify the effects of a singular interaction on the time evolution of a large system of interacting quantum indistinguishable particles.
The project will impact our understanding of various systems of practical interest where the analytic techniques of homogenization theory, spectral analysis, differential equations as well as probabilistic methods, e.g., stochastic analysis, extreme order statistics and theory of disordered systems, etc., play an important role. A number of projects are devised to facilitate training, and inclusion in research, of postdocs and graduate students who have interest in Probability Theory and Mathematical Physics.
Many physical materials and systems, no matter how pure they may appear at human scale, are macroscopically tremendously varied. It is thus an important goal for science to link the known facts about the microscopic structure to overal material properties. A principal goal of the project is to understand this link using tools of mathematics. The concept of a scaling limit is one that plays quite an important role in this endeavor. As is common in mathematics and theoretical sciences in general, the project addresses the above goal by analyzing in full detail a number of specific model-cases of interest. These either stem directly from applied fields (physics and chemistry) or have been processed by other mathematicians and identified as the next key obstacles to tackle. The specific problems that have been addressed in the project include various questions in disordered systems theory such as long-time behavior of random walks in random environment and fluctuation theory of random resistor networks. Further, the project also addresses the subtleties of phase transitions and also fluctuation statistics of various random fields. In each of these problems, a precise mathematical question is formulated and its solution (that is, a proof of a theorem) then presented in detail. The findings made under the auspices of this project advance the knowledge of probability theory and its particular aspects related to disordered systems, phase coexistence and limit theory for random walks and random fields. These have a direct bearing on other subjects of mathematics (e.g., homogenization theory of semi-elliptic differential equations, spectra of random operators) as well as other disciplines of science (for instance, statistical mechanics in physics). In terms of broader impacts, much of the work done in this project is carried out in close collaboration with graduate students, postdocs and other junior scientists. The project thus directly contributes to workforce training in mathematics and academic excellence in general.
