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

Cheliotis, Dimitris; Chino, Yuki; Poisat, Julien

Annales Henri Lebesgue, Tome 2 (2019), pp. 281-329.

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 $\widehat{\alpha }>0$ 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 $\widehat{\alpha }>2$ (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 $\widehat{\alpha }\in (1,2)$ (corrélations non sommables) reste en grande partie ouvert, même si nous prouvons que dans ce cas le désordre est pertinent pour $\alpha >1/\widehat{\alpha }$, une condition que l’on suppose non optimale. Nous donnons des prédictions quant au critère précis de pertinence. Enfin, nous traitons le cas $\widehat{\alpha }\in (0,1)$, 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.

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 $\widehat{\alpha }>0$. 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 $\widehat{\alpha }>2$ (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 $\widehat{\alpha }\in (1,2)$ (non-summable correlations) remains largely open, but we are able to prove that disorder is relevant for $\alpha >1/\widehat{\alpha }$, a condition that is expected to be non-optimal. Predictions on the criterion for disorder relevance in this case are discussed. Finally, the case $\widehat{\alpha }\in (0,1)$ 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.

Classification : 82B44, 82B27, 82D60, 60K05, 60K35
Mots clés : Pinning model, localization transition, free energy, correlated disorder, renewal, disorder relevance, Harris criterion, smoothing inequality, second moment.

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