2007/01/31 by Fabio Lucio Toninelli
Mathematics · Physics and Astronomy · #Markov Chains and Monte Carlo Methods #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #cond-mat.dis-nn #math-ph #math.MP
paper · pdf · doi:10.1007/s00220-008-0469-6
published as Commun. Math. Phys. 280, 389-401 (2008) · 12 pages; v2: added Theorem 2.6, typos corrected
arxiv created 2007/10/12 · openalex publication_date 2008/03/27 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/29
We consider a renewal process τ=τ0,τ1,... on the integers, where the law of τi-τi-1 has a power-like tail P(τi-τi-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 tau. 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 τoccupies a finite fraction of N to a delocalized phase where the density of τvanishes. In absence of disorder the transition is of first order for α>1 and of higher order for α<1. Moreover, for αranging from 1 to 0, the transition ranges from first to infinite order. Presence of even an arbitrarily small amount of disorder is known to modify the order of transition as soon as α>1/2. 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 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.