Scaling Limits of a Weakly Perturbed Random Interface Model

TL;DR

This paper studies a perturbed random interface model on a discrete torus, revealing its hydrodynamic behavior as a combination of viscous Burgers and heat equations, and characterizing its equilibrium fluctuations as an Ornstein-Uhlenbeck process.

Contribution

It introduces a weak global perturbation to a classical interface model and analyzes its hydrodynamic limits and fluctuation behavior, providing new insights into the effects of asymmetry.

Findings

Hydrodynamic limit combines viscous Burgers and heat equations.

Equilibrium fluctuations converge to an Ornstein-Uhlenbeck process.

Results hold for specific parameter regimes, including prime-sized systems.

Abstract

We consider a random interface model on the discrete torus with $2n$ sites, obtained from the classical corner flip dynamics but with a weak global perturbation, namely an asymmetry of order $n^{-\gamma}$ of the direction of growth that switches direction based on the sign of the total area under the interface. The slopes of this model can be viewed as a non-simple exclusion process at half filling with globally dependent rates. We show that, for $\gamma=1$, the hydrodynamic equation of the empirical density is given by a time concatenation of the viscous Burgers equation and the heat equation. Moreover, for $n$ prime and $\gamma>\frac{6}{7}$, we establish convergence in law of the equilibrium fluctuations to an infinite-dimensional Ornstein-Uhlenbeck process.