Limits of 3D Ising and Solid-On-Solid Interfaces

New York University, New York NY

Investigators

Abstract

This project aims to study a variety of problems addressing fundamental features of the classical three-dimensional Ising model and its (2+1)-dimensional Solid-On-Solid (SOS) approximation at low temperature, which are central to understanding crystal growth and formation. The main focus will be on longstanding conjectures concerning the fluctuations of these surfaces and the scaling limits of the models when positioned on a slope and on a hard wall. This project also provides research opportunities for graduate students. The first research direction concerns the 3D Ising model and its (2+1)D SOS approximation at low temperature, with boundary conditions that force these random surfaces to have a positive slope. It is widely believed that, above a slope, the low temperature rigidity of these surfaces will destabilize, and their scaling limits should be a Gaussian Free Field (GFF). Little progress has been made on this problem until a recent successful study of a variant of the SOS model. The project outlines a program building on this recent progress, aiming to establish a GFF limit for the true SOS model and, thereafter, 3D Ising. The second research direction focuses on the 3D Ising and (2+1)D SOS surfaces above a hard flat wall. The plan is to solve key questions, addressing the typical height and shape of the Ising surface in that setting, and the fluctuations of the SOS level lines. The third research direction aims to study the SOS model above a hard wall (as above) but with an added pinning/layering potential, competing with entropic repulsion effects. The plan is to study the surface shape, dynamical phase transitions, and metastability at and near the associated critical points. 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.