Hierarchical pinning models, quadratic maps and quenched disorder

Abstract

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 1/2<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 is larger or equal to 1/2, but not if alpha is smaller than 1/2. Our main result is a proof of these conjectures for the case alpha different from 1/2. We emphasize that for alpha>1/2 we find the correct scaling form (for weak disorder) of the critical point shift.

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

References in corpus (8)

• Fractional moment bounds and disorder relevance for pinning models
• Smoothing effect of quenched disorder on polymer depinning transitions
• Marginal relevance of disorder for pinning models
• A replica-coupling approach to disordered pinning models
• Disordered pinning models and copolymers: beyond annealed bounds
• Estimates on path delocalization for copolymers at selective interfaces
• Critical points of quadratic renormalizations of random variables and phase transitions of disordered polymer models on diamond lattices
• On the irrelevant disorder regime of pinning models

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• An exactly solvable continuous-time Derrida--Retaux model
• The free energy in the Derrida--Retaux recursive model
• Log-periodic critical amplitudes: a perturbative approach
• Oscillatory critical amplitudes in hierarchical models and the tail of the Harris random variable
• The critical behaviors and the scaling functions of a coalescence equation
• Continuum directed random polymers on disordered hierarchical diamond lattices
• The stochastic heat equation with multiplicative Lévy noise: Existence, moments, and intermittency
• Results and conjectures on a toy model of depinning
• Non-coincidence of Quenched and Annealed Connective Constants on the supercritical planar percolation cluster
• Coarse graining, fractional moments and the critical slope of random copolymers