Sample path large deviations for Laplacian models in $(1+1)$-dimensions

Abstract

We study scaling limits of a Laplacian pinning model in (1 + 1) dimension and derive sample path large deviations for the profile height function. The model is given by a Gaussian integrated random walk (or a Gaussian integrated random walk bridge) perturbed by an attractive force towards the zero-level. We study in detail the behaviour of the rate function and show that it can admit up to five minimisers depending on the choices of pinning strength and boundary conditions. This study complements corresponding large deviation results for Gaussian gradient systems with pinning in (1 + 1)-dimension ([FS04]) in (1 + d)-dimension ([BFO09]), and recently in higher dimensions in [BCF14].where ∆ denotes the discrete Laplacian, ∆φ k = φ k+1 + φ k−1 − 2φ k . Our pinning models are then given by the probability measures(εδ 0 (dφ k ) + dφ k ) k∈{−1,0,N,N +1}

The scaling limit of the membrane model

“…Nonetheless it is possible, via analytic and numerical methods, to obtain sharp results on its behaviour. Examples are the study of the entropic repulsion and pinning effects (Adams et al, 2016, Bolthausen et al, 2017, Caravenna and Deuschel, 2008, Kurt, 2007, 2009, extreme value theory (Chiarini et al, 2016), and connections to other statistical mechanics models . In this framework we present our work which aims at determining the scaling limit of the bilaplacian model.…”

Localization of a Gaussian membrane model with weak pinning potentials

“…In the one-dimensional case, the membrane model corresponds to the law of an integrated random walk and a renewal-type argument works well. In particular, its scaling limits have been studied in detail (see Adams et al, 2016;Caravenna and Deuschel, 2009).…”

Exponential Decay of Covariances for the Supercritical Membrane Model

“…If V is convex, there is still a random walk representation of the correlation, the Helffer-Sjöstrand representation, but in the case of non-convex V, random walk techniques cannot be applied, and many of the very basic questions are still open. For a recent investigation, see Adams (2006), Adams et al (2016). The so-called massive free field has the Hamiltonian…”

“…Results on the membrane model with pinning were shown in (1 + 1) dimensions by Caravenna and Deuschel (2008) using a renewal type of argument which, however, is not applicable in higher dimensions. We would like to mention also the work Adams et al (2016) on large deviation principles under a Laplacian interaction without using renewal type arguments. Structure of the paper.…”