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Enhanced interface repulsion from quenched hard–wall randomness
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  • Published: December 2002

Enhanced interface repulsion from quenched hard–wall randomness

  • Daniela Bertacchi1 &
  • Giambattista Giacomin2 

Probability Theory and Related Fields volume 124, pages 487–516 (2002)Cite this article

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Abstract.

 We consider the harmonic crystal, or massless free field, , , that is the centered Gaussian field with covariance given by the Green function of the simple random walk on ℤ d. Our main aim is to obtain quantitative information on the repulsion phenomenon that arises when we condition to be larger than , is an IID field (which is also independent of ϕ), for every x in a large region , with N a positive integer and D a bounded subset of ℝ d. We are mostly motivated by results for given typical realizations of σ (quenched set–up), since the conditioned harmonic crystal may be seen as a model for an equilibrium interface, living in a (d+1)–dimensional space, constrained not to go below an inhomogeneous substrate that acts as a hard wall. We consider various types of substrate and we observe that the interface is pushed away from the wall much more than in the case of a flat wall as soon as the upward tail of σ 0 is heavier than Gaussian, while essentially no effect is observed if the tail is sub–Gaussian. In the critical case, that is the one of approximately Gaussian tail, the interplay of the two sources of randomness, ϕ and σ, leads to an enhanced repulsion effect of additive type. This generalizes work done in the case of a flat wall and also in our case the crucial estimates are optimal Large Deviation type asymptotics as of the probability that ϕ lies above σ in D N .

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Authors and Affiliations

  1. Università di Milano–Bicocca Dipartimento di Matematica e Applicazioni, Via Bicocca degli Arcimboldi 8, 20126 Milano, Italy. e-mail: bertacchi@matapp.unimib.it, , , , , , IT

    Daniela Bertacchi

  2. Université Paris 7 and Laboratoire de Probabilités et Modèles Aléatoires C.N.R.S. UMR 7599, U.F.R. Mathématiques, Case 7012, 2 Place Jussieu, F-75251 Paris, France. home page: http://felix.proba.jussieu.fr/pageperso/giacomin/GBpage.html e-mail: giacomin@math.jussieu.fr, , , , , , FR

    Giambattista Giacomin

Authors
  1. Daniela Bertacchi
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  2. Giambattista Giacomin
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Additional information

Received: 6 February 2002 / Revised version: 23 May 2002 / Published online: 30 September 2002

Mathematics Subject Classification (2000): 82B24, 60K35, 60G15

Keywords or phrases: Harmonic Crystal – Rough Substrate – Quenched and Annealed Models – Entropic Repulsion – Gaussian fields – Extrema of Random Fields – Large Deviations – Random Walks

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Bertacchi, D., Giacomin, G. Enhanced interface repulsion from quenched hard–wall randomness. Probab Theory Relat Fields 124, 487–516 (2002). https://doi.org/10.1007/s004400200223

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  • Issue Date: December 2002

  • DOI: https://doi.org/10.1007/s004400200223

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Keywords

  • Free Field
  • Bounded Subset
  • Simple Random Walk
  • Hard Wall
  • Gaussian Field
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