Université Paris 6Pierre et Marie Curie Université Paris 7Denis Diderot CNRS U.M.R. 7599 Probabilités et Modèles Aléatoires''

### Enhanced interface repulsion from quenched hard--wall randomness

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