Pinning on a defect line: characterization of marginal disorder relevance and sharp asymptotics for the critical point shift
Berger, Quentin; Lacoin, Hubert (2018), Pinning on a defect line: characterization of marginal disorder relevance and sharp asymptotics for the critical point shift, Journal de l'Institut de Mathématiques de Jussieu, 17, 2, p. 305-346. 10.1017/S1474748015000481
TypeArticle accepté pour publication ou publié
Journal nameJournal de l'Institut de Mathématiques de Jussieu
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Abstract (EN)The effect of disorder for pinning models is a subject which has attracted much attention in theoretical physics and rigorous mathematical physics. A peculiar point of interest is the question of coincidence of the quenched and annealed critical point for a small amount of disorder. The question has been mathematically settled in most cases in the last few years, giving in particular a rigorous validation of the Harris Criterion on disorder relevance. However, the marginal case, where the return probability exponent is equal to 1/2, i.e. where the inter-arrival law of the renewal process is given by K(n)=n−3/2ϕ(n) where ϕ is a slowly varying function, has been left partially open. In this paper, we give a complete answer to the question by proving a simple necessary and sufficient criterion on the return probability for disorder relevance, which confirms earlier predictions from the literature. Moreover, we also provide sharp asymptotics on the critical point shift: in the case of the pinning (or wetting) of a one dimensional simple random walk, the shift of the critical point satisfies the following high temperature asymptoticslimβ→0β2loghc(β)=−π2.This gives a rigorous proof to a claim of B. Derrida, V. Hakim and J. Vannimenus (Journal of Statistical Physics, 1992).
Subjects / KeywordsHarris Criterion; Disordered Pinning/Wetting Model; Localization Transition; Disorder Relevance
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