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Sample path large deviations for Laplacian models in \((1+1)\)-dimensions. (English) Zbl 1354.60024

Summary: 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 [T. Funaki and H. Sakagawa, Adv. Stud. Pure Math. 39, 173–211 (2004; Zbl 1221.60138)] in \((1+d) \)-dimension [E. Bolthausen et al., Probab. Theory Relat. Fields 143, No. 3–4, 441–480 (2009; Zbl 1180.60077)], and recently in higher dimensions in [E. Bolthausen et al., J. Math. Soc. Japan 67, No. 4, 1359–1412 (2015; Zbl 1334.60205)].

MSC:

60F10 Large deviations
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60G50 Sums of independent random variables; random walks
60G17 Sample path properties
60G15 Gaussian processes
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
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