Hierarchical pinning models, quadratic maps and quenched disorder

Details Hierarchical pinning models, quadratic maps and quenched disorder Hierarchical pinning models, quadratic maps and quenched disorder

2007-11-29

arxiv.org/abs/0711.4649v3

Probab. Theory Rel. Fields 147, 185-216 (2010)

Mathematics

Probability

26 pages, 2 figures. v3: Theorem 1.6 improved. To appear on Probab. Theory Rel. Fields

Scientific paper

10.1007/s00440-009-0205-y

We consider a hierarchical model of polymer pinning in presence of quenched disorder, introduced by B. Derrida, V. Hakim and J. Vannimenius in 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 alpha>0, 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 by Derrida et al. (1992) that disorder is relevant (respectively, irrelevant or marginally relevant) if 1/21/2 we find the correct scaling form (for weak disorder) of the critical point shift.

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