Localization of a Gaussian membrane model with weak pinning potentials

Abstract

We consider a class of effective models on Z d called Gaussian membrane models with square-well pinning or δ-pinning. It is known that when d = 1 this model exhibits a localization/delocalization transition that depends on the strength of the pinning. In this paper, we show that when d ≥ 2, once we impose weak pinning potentials the field is always localized in the sense that the corresponding free energy is always positive. We also discuss the case that both square-well potentials and repulsive potentials are acting in high dimensions.

Pinning for the critical and supercritical membrane model

“…It turns out that the answer is yes in dimension 1, and no in d ≥ 2. More precisely, if d ≥ 2 or d = 1 and ε > ε c for some ε c > 0 the expected fraction of points in Λ that are pinned is bounded below by a constant (uniformly in Λ) [CD08,Sak12,Sak18].…”

Scaling limit of semiflexible polymers: a phase transition

“…We refer to Caravenna and Deuschel (2009), Cipriani et al (2018b), Hryniv and Velenik (2009) for the scaling limit of the membrane model in d ≥ 1. The literature on the case when κ 1 > 0, κ 2 > 0 is limited and has been considered in the works of Borecki (2010), Borecki and Caravenna (2010), Cipriani et al (2018a), Sakagawa (2018). Borecki (2010) and Borecki and Caravenna (2010) introduced this model as the (∇ + ∆)model (we will also refer to it as "mixed model") with constant κ 1 , κ 2 .…”

“…They studied in d = 1 the influence of pinning in order to understand the localization behavior of the polymer. The results were extended to higher dimensions, together with further properties of the free energy, in Sakagawa (2018). In Cipriani et al (2018a) the scaling limit of the (∇ + ∆)-model is studied.…”