Enhanced interface repulsion from quenched hard-wall randomness

TL;DR

This paper investigates how a harmonic crystal interface in high dimensions is repelled by a quenched random wall, revealing that the tail behavior of the wall's distribution significantly influences the extent of repulsion.

Contribution

It provides a quantitative analysis of interface repulsion in a harmonic crystal conditioned on a quenched random substrate, highlighting the impact of tail behavior on the phenomenon.

Findings

Heavier-than-Gaussian tails cause stronger interface repulsion.

Sub-Gaussian tails result in negligible repulsion effects.

Gaussian tail case shows an additive enhancement of repulsion due to combined randomness.

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.