The evaluation of probabilities of rare events leads to an understanding of the mechanisms by which they can occur, and when many different rare events are possible, it identifies the mechanism causing one of them to actually occur. This approach has been very successful, often for finding the quenched asymptotic behavior when two levels of randomness are present, conditioned on a fixed, typical realization of one element of randomness. The proposed activity focuses on the applications of this approach in two directions:
1. Experiments and simulations involving dynamical systems out of equilibrium report an aging phenomenon, that is "memory" being accumulated in the system. The fluctuation-dissipation theorem of statistical physics relates the time derivative of the correlation function at equilibrium to the effect of small perturbations in the dynamical equations. It is predicted in the physics literature that aging is related to a specific form of breakup of the fluctuation-dissipation theorem. One goal of the proposed activity is to form the mathematical version of the fluctuation-dissipation theorem, to understand the causes of the aging mechanism, and to rigorously relate the two.
2. The Gaussian free field and the Brownian motion are two fundamental random objects of great intrinsic beauty and much interest in probability theory (as well as in mathematical physics). The PI and his collaborators have been very successful in studying the fractal geometry of the extremal points of the occupation measure and covering process associated with planar Brownian motion and random walks. Recent works by the PI's students, their collaborators, and others, suggest strong similarity between the extremes of the Gaussian free field and those of the Brownian motion, as well as novel relations between contour lines of random height functions, those of the Gaussian free field and certain conformally invariant loop ensembles. Pursuing further this similarity, one goal of the proposed activity is to gain new insights and develop novel connections between the planar Gaussian free field and Brownian motion.
Over the past three decades, the physics community has developed sophisticated non-rigorous techniques for making very accurate predictions about the asymptotics of random dynamics, Gibbs measures, and planar objects with conformal symmetries. More recently, mathematicians started making significant progress in developing the corresponding rigorous theories and proving some of these predictions. One of the key tools in this process has been the theory of large deviations and the intuition behind it. The proposed activity is partially motivated by such non-rigorous physics predictions. As such, it is expected to result in new mathematical ideas and techniques that, beyond the immediate resolution of challenging problems of much interest to probabilists, would have an impact on the interface between probability theory and statistical physics. This grant will provide graduate training of students in probability theory. The techniques to be developed in this proposal are also expected to have future applications to the theory and practice of universal non-linear filtering, a problem of relevance and much interest for communication theory (see the URL www-stat.stanford.edu/~amir/ for some related activity and preprints).
