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Hierarchical pinning models, quadratic maps and quenched disorder
Authors:Giambattista Giacomin  Hubert Lacoin  Fabio Lucio Toninelli
Institution:1. U.F.R. Mathématiques et Laboratoire de Probabilités et Modèles Aléatoires, Université Paris Diderot (Paris 7), Case 7012 (site Chevaleret), 75205, Paris, France
2. CNRS and Laboratoire de Physique, ENS Lyon, 46 Allée d’Italie, 69364, Lyon, France
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 ? (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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