Cheliotis, Dimitris
Chino, Yuki
Poisat, Julien
The random pinning model with correlated disorder given by a renewal set
Annales Henri Lebesgue, Volume 2 (2019) , pp. 281-329.


KeywordsPinning model, localization transition, free energy, correlated disorder, renewal, disorder relevance, Harris criterion, smoothing inequality, second moment.


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/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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