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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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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DOI: https://doi.org/10.1007/s00440-009-0205-y
Keywords
- Hierarchical models
- Quadratic recurrence equations
- Pinning models
- Disorder
- Harris criterion
Mathematics Subject Classification (2000)
- 60K35
- 82B44
- 37H10