Structure and Evolution of Low Temperature Spin Systems: Entropic Repulsion and Metastability

Spin systems are mathematical models for ferromagnetism and other properties of materials that are governed by interactions of nuclear magnetic moments (spins). This project aims to study a variety of problems addressing fundamental features of some of the most canonical interacting spin systems at low temperature. Specific models to be studied include the three dimensional Ising model (one of the most fundamental models in statistical mechanics) and the (2+1) dimensional Solid-On-Solid model.

One feature that these models exhibit is the entropic repulsion, where an initially flat surface goes through a series of meta-stable states as it builds its height towards equilibrium. The main focus of this project is to study the entropic repulsion phenomenon for these models and its interplay with Glauber dynamics (the natural stochastic process that models the evolution of the system).

To advance understanding of these problems the project will develop new methods in probability theory which would find applications in other studies of interacting spins systems. The project provides research training opportunities for graduate students.

The first research project focuses on the 3D Ising model at low temperature on an infinite cylinder, with mixed boundary conditions—minus above height 0 and plus elsewhere. These give rise rise to an interface, a random surface separating the plus/minus phases, and the PI aims to study the entropic repulsion effect conditioned on this interface being positive.

The main goal is to show that, in order to gain entropy, the surface gradually rises, and its level lines eventually form a single plateau with an a.s. scaling limit given by a Wulff shape; the level line exhibit cube-root fluctuations away from the boundary; and started at a flat initial state, the evolution of the surface towards equilibrium goes through a sequence of meta-stable plateaus, each with a waiting time doubly-exponential in its height. A related research direction aims to study a class of crystal models that approximate 3D Ising, and show that the scaling limit in terms of a Wulff shape and cube-root fluctuations are universal for that entire class.

For the (2+1)D Solid-On-Solid, the goal is to refine our understanding and obtain the order of the fluctuations in the top level line. The final topic to be studied concerns crystal models such as Solid-On-Solid at low temperature when the boundary condition is tilted, which is known to cause the (otherwise flat) surface to become de-localized. Here the goal is to give quantitative bounds on the height fluctuations, and show that Glauber dynamics is no longer exponentially slow due to meta-stable states.

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.