Large Deviations of Interfaces in 3D Potts Models

Abstract

In this thesis, we study the interfaces of the low temperature Potts models on a 3D square lattice. We provide a detailed understanding of the large deviations and extrema of these interfaces, and address fundamental questions regarding the behavior of these interfaces in the presence of a hard floor.

The Potts model on a graph G = (V, E) is a random assignment of colors to vertices of V that penalizes adjacent vertices assigned with different colors. The number of possible colors is given by the integer parameter q ≥ 2, and the aforementioned penalization is governed by the parameter β > 0, the inverse-temperature of the system. Since its introduction in the 1950's, the model has been a rich source of interesting mathematical phenomenon, with strong ties to many other models in statistical mechanics. In particular, the Potts model is a generalization of the Ising model (the q = 2 case, where colors are called spins, and take values {±1}), and can be coupled together with the random-cluster (FK) model.

We are interested in studying the interface between two coexisting phases in the Potts model. The simplest way to exhibit this phenomenon is to take a square cylinder with side length n and impose blue boundary conditions on the bottom half and red boundary conditions on the top half. The interface is then the surface separating the red region from the blue region. In the Ising case, the interface is known to be rigid, and recent advancements in the last decade have yielded the tightness of the maximum of the Ising interface around c_β log n, where c_β is explicitly related to a large deviation rate of the model. We extend these results to the Potts model, where we find that the presence of additional colors demands the study of two interfaces -- the interface between red and non-red colors, I_red, and the interface between blue and non-blue colors, I_blue. We prove that the rates governing the extrema of these interfaces are different, implying an up-down asymmetry. We also prove the analogous results in the random-cluster setting.

We next study these interfaces in the presence of a hard floor. Here we take an n × n × n box with blue boundary conditions on its bottom side and red boundary conditions on its other five sides. Comparing to the cylinder setting from before, there is now an entire slab of blue vertices located at the boundary change from red to blue, which acts as a floor that the interface cannot penetrate. In the Ising case, it is known that this floor repels the interface and causes its typical height above the center to diverge, a phenomenon called entropic repulsion. We prove that this behavior holds also in the Potts model. Moreover, we establish a logarithmically diverging lower bound on the typical interface height, which was not previously known even for the simpler Ising case. This is complemented by a conjecturally sharp upper bound of ⌊ξ⁻¹ log n⌋ where ξ is the same rate function governing the minimum of I_red in the cylinder setting. We then prove that this is the same rate function for a point-to-plane non-red connection under the infinite volume red measure. Establishing a matching lower bound in the above setting remains an interesting open question. To gain insight on this, we turn to the (2+1)D SOS model above a floor. This model has been well studied as a height function approximation to the 3D Ising interface. Our analysis indicates that the effect of the hard floor in the Ising/Potts case is similar to the effect of a pinning potential λ in the SOS model. As λ varies, the model exhibits a localization-delocalization transition about a critical λ_w. We prove that at criticality λ = λ_w, there is delocalization, with rigidity at height ⌊(1 / 6β) log n + 1/3⌋.