A second note on the discrete Gaussian Free Field with disordered pinning on Z^d, d\geq 2

TL;DR

This paper investigates the discrete Gaussian Free Field with disordered pinning on Z^d, demonstrating the existence of a phase where the field is localized despite an overall repulsive environment, using Bernoulli disorder.

Contribution

It establishes that the quenched critical line lies below h=0 in the (b,h) plane for Bernoulli disorder, revealing a non-trivial localized phase.

Findings

The quenched free energy is strictly less than the annealed free energy when positive.

The critical line separating localized and delocalized phases is below h=0.

Existence of a localized phase where the field is repulsed on average.

Abstract

We study the discrete massless Gaussian Free Field on Z^d, d \geq 2, in the presence of a disordered square-well potential supported on a finite strip around zero. The disorder is introduced by reward/penalty interaction coefficients, which are given by i.i.d. random variables. In the previous note, we proved under minimal assumptions on the law of the environment, that the quenched free energy associated to this model exists in R^+, is deterministic, and strictly smaller than the annealed free energy whenever the latter is strictly positive. Here we consider Bernoulli reward/penalty coefficients b e_x + h with P(e_x=-1)=P(e_x=+1)=1/2 for all x in Z^d, and b > 0, h in R. We prove that in the plane (b,h), the quenched critical line (separating the phases of positive and zero free energy) lies strictly below the line h = 0, showing in particular that there exists a non trivial region where the field is localized though repulsed on average by the environment.