Pinning and disorder relevance for the lattice Gaussian Free Field II: the two dimensional case

TL;DR

This paper investigates how disorder affects the phase transition of a two-dimensional lattice Gaussian Free Field interacting with a disordered substrate, showing the critical point remains unchanged and the transition is of infinite order.

Contribution

It proves that in 2D, the critical point is unaffected by disorder and characterizes the infinite order nature of the phase transition.

Findings

Critical point $h_c(eta)$ equals the annealed model's value.

Phase transition is of infinite order in 2D.

Behavior near critical point differs from higher dimensions.

Abstract

This paper continues a study initiated in [34], on the localization transition of a lattice free field on $\mathbb Z^d$ interacting with a quenched disordered substrate that acts on the interface when its height is close to zero. The substrate has the tendency to localize or repel the interface at different sites. A transition takes place when the average pinning potential $h$ goes past a threshold $h_c$: from a delocalized phase $h<h_c$, where the field is macroscopically repelled by the substrate to a localized one $h>h_c$ where the field sticks to the substrate. Our goal is to investigate the effect of the presence of disorder on this phase transition. We focus on the two dimensional case $(d=2)$ for which we had obtained so far only limited results. We prove that the value of $h_c(\beta)$ is the same as for the annealed model, for all values of $\beta$ and that in a neighborhood of $h_c$. Moreover we prove that in contrast with the case $d\ge 3$ where the free energy has a quadratic behavior near the critical point, the phase transition is of infinite order $$\lim_{u\to 0+} \frac{ \log \mathrm{F}(\beta,h_c(\beta)+u)}{(\log u)}= \infty.$$