Scaling limits and phase transitions in spatial random processes

This project aims to advance mathematical theories of phase transitions and critical phenomena in a number of specific contexts. While these effects are ubiquitous in physics and chemistry, understanding them mathematically is challenging due to the need to consider the limit of large systems whose constituents exhibit persistent dependence throughout all scales. Probability theory offers a number of tools and proof strategies for this purpose provided the problem is cast, or can otherwise be represented, via elementary probabilistic concepts such as random walks or random fields. The goal of the analysis is to show that, in several settings, as the system size increases, a new structure called the scaling limit appears. Often quantitative details of the scaling limit are often independent of the specifics of the original problem, demonstrating universality. . Through the specific examples to be studied, new approaches and techniques will be developed that will advance our understanding of scaling limits and phase transitions. The questions are of varied difficulty, which will make it possible to include graduate students and postdocs in this research. The project explores a diverse list of specific questions from probability, mathematical physics and analysis. First, extremal processes of branching Markov chains, local time of random walks and other random fields exhibiting logarithmic correlations will be studied with the aim to establish broad universality of the Gaussian Free Field and develop novel techniques for dealing with these problems. Second, scaling limits of random walks in random environments will be analyzed in examples that test the limits of the existing theory of stochastic homogenization. Here the particular emphasis is on the situations where the underlying environment has good spatial mixing properties and yet is not necessarily drawn from a stationary probability law. Third, a recent treatment of Bose-Einstein condensation in mean-field interacting bosonic gases using Feynman’s random-cycle representation will be expanded to establish a quantum phase transition. The point is to provide a direct link between the properties of the quantum system and those of its graphical representation. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.