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Hierarchical pinning models, quadratic maps and quenched disorder, arXiv:0711.4649 [math.PR

By Giambattista Giacomin, Hubert Lacoin, Fabio and Lucio Toninelli


Abstract. We consider a hierarchical model of polymer pinning in presence of quenched disorder, introduced by B. Derrida, V. Hakim and J. Vannimenus [10], which can be reinterpreted 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 {Rn}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 Rn, 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 R0 is larger than a certain threshold value, and it is zero otherwise. It was conjectured in [10] 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

Topics: Hierarchical Models, Quadratic Recurrence Equations, Pinning Models, Disorder, Harris Criterion
Year: 2013
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