L. R. G. Fontes, D. P. Medeiros, M. Vachkovskaia
Time fluctuations of the random average process
with parabolic initial conditions
(69K, latex)
ABSTRACT. 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 $n^{1-d/4}$
for $d=1,2,3$; $\sqrt{\log n}$ in $d=4$; and are bounded in $d\geq5$.
We establish a central limit theorem in $d=1$. In the bounded case of
$d\geq5$, 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.