1. Enhanced interface repulsion from quenched hard–wall randomness
- Author
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Daniela Bertacchi, Giambattista Giacomin, Bertacchi, D, and Giacomin, G
- Subjects
Statistics and Probability ,large deviation ,Field (physics) ,Gaussian ,FOS: Physical sciences ,Geometry ,82B24 ,60K35 ,60G15 ,quenched and annealed model ,harmonic crystal ,Upper and lower bounds ,rough substrate ,symbols.namesake ,entropic repulsion ,Probability theory ,FOS: Mathematics ,extrema of random field ,Mathematical Physics ,Randomness ,Mathematics ,Gaussian field ,Random field ,Probability (math.PR) ,Mathematical analysis ,random walks ,Mathematical Physics (math-ph) ,Random walk ,MAT/06 - PROBABILITA E STATISTICA MATEMATICA ,Bounded function ,symbols ,Statistics, Probability and Uncertainty ,Mathematics - Probability ,Analysis - Abstract
We consider the harmonic crystal on the d-dimensional lattice, d larger or equal to 3, that is the centered Gaussian field $\phi$ 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 the field to be larger than an IID field $\sigma$ (which is also independent of $\phi$), for every x in a large region $D_N=ND\cap \Z^d$, with N a positive integer and $D \subset\R^d$. We are mostly motivated by results for given typical realizations of the $\sigma$ (quenched set-up), since the conditioned harmonic crystal may be seen as a model for an equilibrium interface, constrained not to go below a 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 much more than in the case of a flat wall as soon as the upward tail of $\sigma$ 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 approximately Gaussian tail, the interplay of the two sources of randomness, $\phi$ and $\sigma$, leads to an enhanced repulsion effect of additive type., Comment: 28 pages
- Published
- 2002