The random pinning model with correlated disorder given by a renewal set
Cheliotis, Dimitris; Chino, Yuki; Poisat, Julien (2019), The random pinning model with correlated disorder given by a renewal set, Annales Henri Lebesgue, 2, p. 281-329
TypeArticle accepté pour publication ou publié
External document linkhttps://hal.archives-ouvertes.fr/hal-01590623v2
Journal nameAnnales Henri Lebesgue
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CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Abstract (EN)We investigate the effect of correlated disorder on the localization transition undergone by a renewal sequence with loop exponent α > 0, when the correlated sequence is given by another independent renewal set with loop exponent α > 0. Using the renewal structure of the disorder sequence, we compute the annealed critical point and exponent. Then, using a smoothing inequality for the quenched free energy and second moment estimates for the quenched partition function, combined with decoupling inequalities, we prove that in the case α > 2 (summable correlations), disorder is irrelevant if α < 1/2 and relevant if α > 1/2, which extends the Harris criterion for independent disorder. The case α ∈ (1, 2) (non-summable correlations) remains largely open, but we are able to prove that disorder is relevant for α > 1/ ˆ α, a condition that is expected to be non-optimal. Predictions on the criterion for disorder relevance in this case are discussed. Finally, the case α ∈ (0, 1) is somewhat special but treated for completeness: in this case, disorder has no effect on the quenched free energy, but the annealed model exhibits a phase transition.
Subjects / KeywordsPinning model; localization transition; free energy; correlated disorder; renewal; disorder relevance; Harris criterion; smoothing inequality; second moment
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