Large-Scale Phenomena in Models of Statistical Mechanics

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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.