Time fluctuations of the random average process with parabolic initial conditions

The random average process is a randomly evolving d-dimensional surface whose heights are updated by random convex combinations of neighboring heights. The fluctuations of this process in case of linear initial conditions have been studied before. In this paper, we analyze the case of polynomial initial conditions of degree 2 and higher. Specifically, we prove that the time fluctuations of a initial parabolic surface are of order n2-d/2 for d=1,2,3; log n in d=4; and are bounded in d[greater-or-equal, slanted]5. We establish a central limit theorem in d=1. In the bounded case of d[greater-or-equal, slanted]5, we exhibit an invariant measure for the process as seen from the average height at the origin and describe its asymptotic space fluctuations. We consider briefly the case of initial polynomial surfaces of higher degree to show that their time fluctuations are not bounded in high dimensions, in contrast with the linear and parabolic cases.