Scaling Limits and Phase Transitions in Spatial Random Processes

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This project is devoted to the study of various aspects of phase transitions and critical phenomena. These originate in physics but the methods of study require significant input from mathematics. A common feature of the problems discussed in the project is that they involve systems of many constituents. It is here where probability theory offers a number of indispensable tools. The specific questions are typically set in the context of a complex microscopic system defined in terms of mathematical concepts such as random walks or random fields. The aim is to show that, as the system size increases, a new structure emerges. This structure, sometimes called a scaling limit, often admits an independent characterization that enables its study using the methods of mathematical analysis. Particular attention is paid to the phenomenon of universality, which refers to independence of the scaling limit to the details of the underlying microscopic system. Understanding universality in its mathematically precise form is one of the overarching goals of the project. The project provides research training opportunities for graduate students. The project is divided roughly into two parts. The first one is focused on the study of extremal problems for spatial random fields. Drawing on earlier success of the analysis of the two-dimensional Gaussian Free Field, the aim is to establish universality of the conclusions for other examples of logarithmically-correlated fields such as the random walk local time or the discrete-Gaussian model below the Kosterlitz-Thouless transition. The interest here is on both the phenomena and the development of techniques required to control such systems. The second part of the project is devoted to the analysis of phase transitions in a number of systems of interest. One of these is a system of interacting bosons, which is proposed to be analyzed using a random-cycle representation. The aim here is to give a rigorous proof of Bose-Einstein condensation at positive temperatures in suitable scaling limits. Another model to be studied is that of interacting random walks; the aim here is to describe the limit shape using a variational problem for the walk range. The project naturally draws on recent advances in the field, some due to the PI. The techniques involved span a number of topics in probability and analysis; for instance, extreme order statistics, homogenization theory, permutation statistics, fixed point theory. A number of problems have been designed with the aim to include graduate students and postdocs in research.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.