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Fluctuations of a surface submitted to
a random average process

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P. A. Ferrari, L. R. G. Fontes

We consider a hyper surface of dimension $d$ imbeded in a $d+1$ space. For
each $x\in\z^d$, 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
combinations 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 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 of the
neighbors 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 of the neighbors 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 uniformly bounded in $t$ in dimensions $d\ge 3$. 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 weak law
of large numbers and 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 lines are independent.

Keywords:
random average process, random surfaces, product of random
matrices, linear process, voter model, smoothing process.
AMS Classification numbers: 60K35, 82C

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