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Hierarchical pinning models, quadratic maps and quenched disorder.
- Source :
-
Probability Theory & Related Fields . May2010, Vol. 147 Issue 1/2, p185-216. 32p. 2 Diagrams. - Publication Year :
- 2010
-
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 R0 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. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 01788051
- Volume :
- 147
- Issue :
- 1/2
- Database :
- Academic Search Index
- Journal :
- Probability Theory & Related Fields
- Publication Type :
- Academic Journal
- Accession number :
- 48148257
- Full Text :
- https://doi.org/10.1007/s00440-009-0205-y