Hierarchical pinning models, quadratic maps and quenched disorder

Abstract

Abstract. We consider a hierarchical model of polymer pinning in presence of quenched disorder, introduced by B. Derrida, V. Hakim and J. Vannimenus [11], which can be reinterpreted 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 {Rn}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 Rn, 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 ∈ (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 [11] 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.

Hierarchical pinning model with site disorder: disorder is marginally relevant

“…We state here all the results on properties of the critical system that have been proved for the bond-disorder model in [9] and that are still true in our framework. Our first result concerns the shape of the free energy curve around zero in the pure system.…”

“…The law P n is the uniform law over all directed path and the path σ (the wall) is marked by a dashed line. In the bond disorder case [6,9] the hierarchy of lattices is the same, but, with reference to D 2 , there would be four variables of disorder this model similar to the random walk based model where the Green function decays with a power law with the length of the system (in the hierarchical context the length of the system is s n ). This is not true for the bond model Green function (every bond is visited with probability b −n ).…”

New Bounds for the Free Energy of Directed Polymers in Dimension 1 + 1 and 1 + 2

“…The various techniques we use have been inspired by ideas used successfully for another polymer model, namely the polymer pinning on a defect line (see [24,14,10,25,15]). However the ideas we use to establish lower bounds differ sensibly from the ones leading to the upper bounds.…”

Pinning and disorder relevance for the lattice Gaussian free field

“…The upper-bound in (2.30) and (2.31) is quite easy to prove and is valid in any dimension. Its proof can be read independently of the rest of the paper: it relies on the disorder tilt and fractional moment bound introduced in [29,21]. However, here the implementation of the idea is remarkably straightforward: no coarse graining procedure is needed (see [28,Section 6] for a review of various coarse graining procedures).…”