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

TL;DR

This paper investigates the effects of disordered pinning on the discrete Gaussian Free Field in multiple dimensions, demonstrating the existence and properties of the quenched free energy under minimal assumptions.

Contribution

It establishes the existence, determinism, and strict inequality of quenched versus annealed free energy for the disordered GFF with minimal environmental assumptions.

Findings

Quenched free energy exists and is deterministic.

Quenched free energy is strictly less than annealed free energy when positive.

Results hold for dimensions d ≥ 2 with minimal assumptions.

Abstract

We study the discrete massless Gaussian Free Field on $\Z^d$, $d\geq2$, 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. Under minimal assumptions on the law of the environment, we prove 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.