Abstract
We consider a renewal process τ = {τ 0, τ 1,...} on the integers, where the law of τ i − τ i-1 has a power-like tail P(τ i − τ i-1 = n) = n −(α+1) L(n) with α ≥ 0 and L(·) slowly varying. We then assign a random, n-dependent reward/penalty to the occurrence of the event that the site n belongs to τ. In such generality this class of problems includes, among others, (1 + d)-dimensional models of pinning of directed polymers on a one-dimensional random defect, (1 + 1)-dimensional models of wetting of disordered substrates, and the Poland-Scheraga model of DNA denaturation. By varying the average of the reward, the system undergoes a transition from a localized phase, where τ occupies a finite fraction of \({\mathbb{N}}\) to a delocalized phase, where the density of τ vanishes. In absence of disorder (i.e., if the reward is independent of n), the transition is of first order for α > 1 and of higher order for α < 1. Moreover, for α ranging from 1 to 0, the transition ranges from first to infinite order. Presence of even an arbitrarily small (but extensive) amount of disorder is known to modify the order of transition as soon as α > 1/2 [11]. In physical terms, disorder is relevant in this situation, in agreement with the heuristic Harris criterion. On the other hand, for 0 < α < 1/2 it has been proven recently by K. Alexander [2] that, if disorder is sufficiently weak, critical exponents are not modified by randomness: disorder is irrelevant. In this work, generalizing techniques which in the framework of spin glasses are known as replica coupling and interpolation, we give a new, simpler proof of the main results of [2]. Moreover, we (partially) justify a small-disorder expansion worked out in [9] for α < 1/2, showing that it provides a free energy upper bound which improves the annealed one.
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Toninelli, F.L. A Replica-Coupling Approach to Disordered Pinning Models. Commun. Math. Phys. 280, 389–401 (2008). https://doi.org/10.1007/s00220-008-0469-6
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DOI: https://doi.org/10.1007/s00220-008-0469-6