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The random pinning model with correlated disorder given by a renewal set

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Date
2019
Link to item file
https://hal.archives-ouvertes.fr/hal-01590623v2
Dewey
Probabilités et mathématiques appliquées
Sujet
Pinning model; localization transition; free energy; correlated disorder; renewal; disorder relevance; Harris criterion; smoothing inequality; second moment
Journal issue
Annales Henri Lebesgue
Number
2
Publication date
07-2019
Article pages
281-329
URI
https://basepub.dauphine.fr/handle/123456789/20371
Collections
  • CEREMADE : Publications
Metadata
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Author
Cheliotis, Dimitris
Chino, Yuki
Poisat, Julien
60 CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Type
Article accepté pour publication ou publié
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.

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