Université Paris 6
Pierre et Marie Curie | Université Paris 7
Denis Diderot | |

CNRS U.M.R. 7599
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``Probabilités et Modèles Aléatoires''
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**Auteur(s): **

**Code(s) de Classification MSC:**

- 82B24 Interface problems
- 60K35 Interacting random processes; statistical mechanics type models; percolation theory, See also {82B43, 82C43}
- 60G15 Gaussian processes

**Résumé:** We consider the {\sl harmonic crystal}, or {\sl massless free field},
$\s=\{\s_x\}_{x\in \Z^d}$, $d\ge 3$, that is the centered Gaussian
field with covariance given by the Green function of the
simple random walk on $\Z^d$.
Our main aim is to obtain quantitative
information on the repulsion phenomenon
that arises when we condition $\s_x$ to be larger than
$\sg_x$, $\sg=\{\sg_x\}_{x\in \Z^d}$ is an IID field (which is also
independent of $\s$),
for every $x$ in a {\sl large} region $D_N=ND\cap \Z^d$, with $N$ a positive
integer and $D$ a bounded
subset of $\R^d$. We are mostly motivated by
results for given typical
realizations of $\sg$ ({\sl quenched} set--up), since the conditioned
harmonic crystal may be seen as a model for an equilibrium interface,
living in a $(d+1)$--dimensional space, constrained not to go below
an inhomogeneous substrate that acts as a hard wall.
We consider various types of substrate and we observe that
the
interface is pushed away from the wall {\sl much}
more than in the case of a flat wall as soon as the
upward tail of $\sg_0$ is heavier than Gaussian, while essentially
no effect is observed if the tail is sub--Gaussian.
In the critical case, that is the one of {\sl approximately Gaussian} tail,
the interplay of the two sources of randomness,
$\s$ and $\sg$, leads to an
enhanced repulsion effect of {\sl additive} type.
This generalizes work done in the case of
a flat wall and also in our case the crucial estimates
are optimal Large Deviation type asymptotics as $N\nearrow \infty$
of the probability that $\s$ lies above $\sg$
in $D_N$.

**Mots Clés:** *Harmonic Crystal ; Rough Substrate ; Quenched and Annealed Models ;
Entropic Repulsion ; Gaussian fields ; Extrema of Random Fields ; Large Deviations ;
Random Walks *

**Date:** 2002-07-10

**Prépublication numéro:** *PMA-750*

**Pdf file : **PMA-750.pdf