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Hierarchical pinning models, quadratic maps and quenched disorder
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  • Published: 26 February 2009

Hierarchical pinning models, quadratic maps and quenched disorder

  • Giambattista Giacomin1,
  • Hubert Lacoin1 &
  • Fabio Lucio Toninelli2 

Probability Theory and Related Fields volume 147, pages 185–216 (2010)Cite this article

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  • 20 Citations

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Abstract

We consider a hierarchical model of polymer pinning in presence of quenched disorder, introduced by Derrida et al. (J Stat Phys 66:1189–1213, 1992), which can be re-interpreted as an infinite dimensional dynamical system with random initial condition (the disorder). It is defined through a recurrence relation for the law of a random variable {R n }n=1,2, ..., which in absence of disorder (i.e., when the initial condition is degenerate) reduces to a particular case of the well-known logistic map. The large-n limit of the sequence of random variables 2−n log R n , a non-random quantity which is naturally interpreted as a free energy, plays a central role in our analysis. The model depends on a parameter α ϵ (0, 1), related to the geometry of the hierarchical lattice, and has a phase transition in the sense that the free energy is positive if the expectation of R 0 is larger than a certain threshold value, and it is zero otherwise. It was conjectured in Derrida et al. (J Stat Phys 66:1189–1213, 1992) that disorder is relevant (respectively, irrelevant or marginally relevant) if 1/2 < α < 1 (respectively, α < 1/2 or α = 1/2), in the sense that an arbitrarily small amount of randomness in the initial condition modifies the critical point with respect to that of the pure (i.e., non-disordered) model if α ≥ 1/2, but not if α <  1/2. Our main result is a proof of these conjectures for the case α ≠ 1/2. We emphasize that for α >  1/2 we find the correct scaling form (for weak disorder) of the critical point shift.

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Authors and Affiliations

  1. U.F.R. Mathématiques et Laboratoire de Probabilités et Modèles Aléatoires, Université Paris Diderot (Paris 7), Case 7012 (site Chevaleret), 75205, Paris, France

    Giambattista Giacomin & Hubert Lacoin

  2. CNRS and Laboratoire de Physique, ENS Lyon, 46 Allée d’Italie, 69364, Lyon, France

    Fabio Lucio Toninelli

Authors
  1. Giambattista Giacomin
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  2. Hubert Lacoin
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  3. Fabio Lucio Toninelli
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Correspondence to Giambattista Giacomin.

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Giacomin, G., Lacoin, H. & Toninelli, F.L. Hierarchical pinning models, quadratic maps and quenched disorder. Probab. Theory Relat. Fields 147, 185–216 (2010). https://doi.org/10.1007/s00440-009-0205-y

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  • Received: 21 December 2007

  • Revised: 02 February 2009

  • Published: 26 February 2009

  • Issue Date: May 2010

  • DOI: https://doi.org/10.1007/s00440-009-0205-y

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Keywords

  • Hierarchical models
  • Quadratic recurrence equations
  • Pinning models
  • Disorder
  • Harris criterion

Mathematics Subject Classification (2000)

  • 60K35
  • 82B44
  • 37H10
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