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

We investigate the effect of correlated disorder on the localization transition undergone by a renewal sequence with loop exponent $\alpha >0$, when the correlated sequence is given by another independent renewal set with loop exponent $\stackrel{^}{\alpha }>0$. 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 $\stackrel{^}{\alpha }>2$ (summable correlations), disorder is irrelevant if $\alpha <1/2$ and relevant if $\alpha >1/2$, which extends the Harris criterion for independent disorder. The case $\stackrel{^}{\alpha }\in \left(1,2\right)$ (non-summable correlations) remains largely open, but we are able to prove that disorder is relevant for $\alpha >1/\stackrel{^}{\alpha }$, a condition that is expected to be non-optimal. Predictions on the criterion for disorder relevance in this case are discussed. Finally, the case $\stackrel{^}{\alpha }\in \left(0,1\right)$ 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.

Nous étudions l’effet d’un désordre corrélé sur la transition d’accrochage pour une suite de renouvellement d’exposant $\alpha >0$, lorsque celui-ci est donné par une suite de renouvellement d’exposant $\stackrel{^}{\alpha }>0$ indépendante de la première. En utilisant la structure de renouvellement du désordre, nous calculons le point et l’exposant critiques annealed. Puis, à l’aide d’une inégalité de lissage et d’estimations sur le deuxième moment de la fonction de partition quenched, ainsi que des inégalités de découplage, nous prouvons que dans le cas $\stackrel{^}{\alpha }>2$ (corrélations sommables) le désordre est non-pertinent pour $\alpha <1/2$ et pertinent si $\alpha >1/2$, ce qui étend le critère de Harris pour un désordre sans corrélation. Le cas $\stackrel{^}{\alpha }\in \left(1,2\right)$ (corrélations non sommables) reste en grande partie ouvert, même si nous prouvons que dans ce cas le désordre est pertinent pour $\alpha >1/\stackrel{^}{\alpha }$, une condition que l’on suppose non optimale. Nous donnons des prédictions quant au critère précis de pertinence. Enfin, nous traitons le cas $\stackrel{^}{\alpha }\in \left(0,1\right)$, bien que particulier, pour compléter l’étude : dans ce cas-là, le désordre n’a aucun effet sur l’énergie libre quenched, mais le modèle annealed présente une transition de phase.

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DOI: 10.5802/ahl.11
Classification: 82B44,  82B27,  82D60,  60K05,  60K35
Keywords: Pinning model, localization transition, free energy, correlated disorder, renewal, disorder relevance, Harris criterion, smoothing inequality, second moment.
Cheliotis, Dimitris 1; Chino, Yuki 2; Poisat, Julien 3

1 National and Kapodistrian University of Athens Department of Mathematics Panepistimiopolis 15784 Athens (Greece)
2 Mathematical Institute Leiden University P.O. Box 9512 2300 RA Leiden (The Netherlands)
3 CEREMADE, CNRS Université Paris-Dauphine, Université PSL 75016 Paris (France)
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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. doi : 10.5802/ahl.11. http://archive.numdam.org/articles/10.5802/ahl.11/

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