Hierarchical pinning model in correlated random environment

Berger, Quentin; Toninelli, Fabio Lucio

Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013) no. 3, pp. 781-816.

Nous considérons le modèle hiérarchique d'accrochage sur une ligne de défaut inhomogène étudié dans (J. Statist. Phys. 66 (1992) 1189-1213), qui possède une transition de phase de localisation/délocalisation. Dans le cas où le désordre est i.i.d. (indépendant et identiquement distribué), la question de pertinence/non pertinence du désordre (i.e. de savoir si le désordre change ou non les propriétés critiques du système par rapport au cas homogène) est maintenant bien comprise d'un point de vue mathématique (Probab. Theory Related Fields 148 (2010) 159-175, Pure Appl. Math. 63 (2010) 233-265). Nous considérons ici le cas où le désordre est corrélé spatialement, et où les corrélations respectent la structure hiérarchique du modèle; dans le cadre non-hiérarchique, notre choix correspondrait à une décroissance en loi de puissance des corrélations. En termes d'exposant critique du modèle homogène et d'exposant de décroissance des corrélations, nous identifions trois régions. Dans la première (corrélations non sommables), la transition de phase disparaît. Dans la deuxième (corrélations décroissant suffisamment vite), le système se comporte essentiellement comme dans le cas i.i.d., et le critère de pertinence/non pertinence du désordre n'est pas modifié. Enfin, il existe une région où le présence de corrélations change les propriétés critiques du système annealed.

We consider the hierarchical disordered pinning model studied in (J. Statist. Phys. 66 (1992) 1189-1213), which exhibits a localization/delocalization phase transition. In the case where the disorder is i.i.d. (independent and identically distributed), the question of relevance/irrelevance of disorder (i.e. whether disorder changes or not the critical properties with respect to the homogeneous case) is by now mathematically rather well understood (Probab. Theory Related Fields 148 (2010) 159-175, Pure Appl. Math. 63 (2010) 233-265). Here we consider the case where randomness is spatially correlated and correlations respect the hierarchical structure of the model; in the non-hierarchical model our choice would correspond to a power-law decay of correlations. In terms of the critical exponent of the homogeneous model and of the correlation decay exponent, we identify three regions. In the first one (non-summable correlations) the phase transition disappears. In the second one (correlations decaying fast enough) the system behaves essentially like in the i.i.d. setting and the relevance/irrelevance criterion is not modified. Finally, there is a region where the presence of correlations changes the critical properties of the annealed system.

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