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arXiv · 1512.05240

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

Hubert Lacoin
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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 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 $h_c(β)$ is the same as for the annealed model, for all values of $β$ 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 $$\lim_{u\to 0+} \frac{ \log \mathrm{F}(β,h_c(β)+u)}{(\log u)}= \infty.$$

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Hubert Lacoin. 2016-02-16. Pinning and disorder relevance for the lattice Gaussian Free Field II: the two dimensional case. https://arxiv.org/abs/1512.05240

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