Fluctuations of a Surface Submitted to a Random Average Process
P.A. Ferrari, Universidade de Säo Paulo
L. R. G. Fontes, Universidade de Säo Paulo
Abstract
We consider a hypersurface of dimension $d$ imbedded in a $d+1$ dimensional
space. For each $xinz^d$, let $eta_t(x)in R$ be the height of the
surface at site $x$ at time $t$. At rate $1$ the $x$-th height is updated to
a random convex combination of the heights of the `neighbors' of $x$. The
distribution of the convex combination is translation invariant and does not
depend on the heights. This motion, named the random average process (RAP),
is one of the linear processes introduced by Liggett (1985). Special
cases of RAP are a type of smoothing process (when the convex combination is
deterministic) and the voter model (when the convex combination concentrates
on one site chosen at random). We start the heights located on a hyperplane
passing through the origin but different from the trivial one $eta(x)equiv
0$. We show that, when the convex combination is neither deterministic nor
concentrating on one site, the variance of the height at the origin at time
$t$ is proportional to the number of returns to the origin of a symmetric
random walk of dimension $d$. Under mild conditions on the distribution of
the random convex combination, this gives variance of the order of $t^{1/2}$
in dimension $d=1$, $log t$ in dimension $d=2$ and bounded in $t$ in
dimensions $dge 3$. We also show that for each initial hyperplane the
process as seen from the height at the origin converges to an invariant
measure on the hyper surfaces conserving the initial asymptotic slope. The
height at the origin satisfies a central limit theorem. To obtain the
results we use a corresponding probabilistic cellular automaton for which
similar results are derived. This automaton corresponds to the product of
(infinitely dimensional) independent random matrices whose rows are
independent.
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