Scaling limit of an equilibrium surface under the Random Average Process

TL;DR

This paper studies the large-scale fluctuations of an equilibrium surface in the Random Average Process, revealing their convergence to Gaussian processes, including Brownian motion in one dimension and a discontinuous covariance process in two dimensions.

Contribution

It provides the first rigorous derivation of the scaling limits of surface fluctuations in the Random Average Process in one and two dimensions.

Findings

Fluctuations in 1D converge to Brownian motion.

In 2D, fluctuations converge to a Gaussian process with discontinuous covariance.

Correlation structure matches that of Discrete Gaussian Free Fields.

Abstract

We consider the equilibrium surface of the Random Average Process started from an inclined plane, as seen from the height of the origin, obtained in [Ferrari & Fontes, 1998], where its fluctuations were shown to be of order of the square root of the distance to the origin in one dimension, and the square root of the log of that distance in two dimensions (and constant in higher dimensions). Remarkably, even if not pointed out explicitly in [Ferrari & Fontes, 1998], the correlation structure of those fluctuations is given in terms of the Green's function of a certain random walk, and thus corresponds to those of Discrete Gaussian Free Fields. In the present paper we obtain the scaling limit of those fluctuations in one and two dimensions, in terms of Gaussian processes, in the sense of finite dimensional distributions. In one dimension, the limit is given by Brownian Motion; in two dimensions, we get a process with a discontinuous covariance function.