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

Lacoin, Hubert

doi:10.24033/asens.2411

Pinning and disorder relevance for the lattice Gaussian Free Field II: The two dimensional case
[Interaction et pertinence du désordre pour le champ libre gaussien sur un réseau II : le cas bi-dimensionnel]

Lacoin, Hubert

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 52 (2019) no. 6, pp. 1331-1401.

Le texte intégral des articles récents est réservé aux abonnés de la revue. Consultez l'article sur le site de la revue.

Résumé
Abstract

Cet article approfondit l'étude (commencée dans [35]) de la transition de localisation pour un champ libre gaussien défini sur le réseau $ℤ^{d}$ en interaction avec un substrat désordonné qui affecte les points situés proches de la hauteur zéro. Le substrat peut avoir un effet attracteur ou répulsif selon le site considéré. Une transition a lieu lorsque le potentiel moyen d'interaction $h$ dépasse un certain seuil $h_{c}$ : cette valeur critique définit une phase délocalisée $h < h_{c}$ , au sein de laquelle le champ est globalement repoussé par le substrat, et une phase localisée $h > h_{c}$ ou le champ adhère au substrat. Notre objectif est d'évaluer les effets de la présence de désordre pour cette transition de phase. Nous nous concentrons sur le cas bi-dimensionnel $(d = 2)$ , et démontrons que la valeur du point critique $h_{c} (β)$ coincide avec celle du modèle moyenné (ou annealed), et ce quelle que soit la valeur de l'intensité du désordre $β$ . De plus, nous démontrons que, contrairement au cas $d \geq 3$ pour lequel l'énergie libre a un comportement quadratique au voisinage du point critique, la transition de phase est ici d'ordre infini

lim_{u \to 0 +} \frac{log f (β, h_{c} (β) + u)}{(log u)} = \infty .

Un résultat analogue est exposé pour le modèle de co-membrane bi-dimensionnelle.

This paper continues a study initiated in [35], on the localization transition of a lattice free field on $ℤ^{d}$ 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 $h_{c}$ : this critical value separates a delocalized phase $h < h_{c}$ , where the field is macroscopically repelled by the substrate from 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 $h_{c} (β)$ is the same as for the annealed model, for all values of the disorder intensity $β$ . Moreover we prove that, in contrast with the case $d \geq 3$ 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 f (β, h_{c} (β) + u)}{(log u)} = \infty .

An analogous result is presented for the two dimensional co-membrane model.

Publié le : 2019-11-18

DOI : 10.24033/asens.2411

Classification : 60K35, 60K37, 82B27, 82B44
Keywords: Lattice Gaussian free field, disordered pinning model, localization transition, critical behavior, disorder relevance, co-membrane model
Mot clés : Champs libre gaussien sur un réseau, modèle d'accrochage désordonné, transition de localisation, comportement critique, pertinence du désordre, modèle de co-membrane

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     title = {Pinning and disorder relevance  for the lattice {Gaussian} {Free} {Field} {II:}  {The} two dimensional case},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
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Lacoin, Hubert. Pinning and disorder relevance  for the lattice Gaussian Free Field II:  The two dimensional case. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 52 (2019) no. 6, pp. 1331-1401. doi : 10.24033/asens.2411. http://www.numdam.org/articles/10.24033/asens.2411/

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