Hierarchical pinning models, quadratic maps and quenched disorder
Résumé
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 alpha I mu (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 < alpha < 1 (respectively, alpha < 1/2 or alpha = 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 alpha a parts per thousand yen 1/2, but not if alpha < 1/2. Our main result is a proof of these conjectures for the case alpha not equal 1/2. We emphasize that for alpha > 1/2 we find the correct scaling form (for weak disorder) of the critical point shift.