Hierarchical pinning model in correlated random environment
Berger, Quentin ; Toninelli, Fabio Lucio
Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013), p. 781-816 / Harvested from Numdam
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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.

Publié le : 2013-01-01
DOI : https://doi.org/10.1214/12-AIHP493
Classification:  82B44,  82D60,  60K37 
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     author = {Berger, Quentin and Toninelli, Fabio Lucio},
     title = {Hierarchical pinning model in correlated random environment},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     volume = {49},
     year = {2013},
     pages = {781-816},
     doi = {10.1214/12-AIHP493},
     mrnumber = {3112434},
     zbl = {1281.82013},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPB_2013__49_3_781_0}
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Berger, Quentin; Toninelli, Fabio Lucio. Hierarchical pinning model in correlated random environment. Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013) pp. 781-816. doi : 10.1214/12-AIHP493. http://gdmltest.u-ga.fr/item/AIHPB_2013__49_3_781_0/

[1] K. S. Alexander. The effect of disorder on polymer depinning transitions. Commun. Math. Phys. 279 (2008) 117-146. | MR 2377630 | Zbl 1175.82034

[2] K. S. Alexander and N. Zygouras. Quenched and annealed critical points in polymer pinning models. Commun. Math. Phys. 291 (2009) 659-689. | MR 2534789 | Zbl 1188.82154

[3] Q. Berger and H. Lacoin. Sharp critical behavior for random pinning model with correlated environment. Stochastic Process. Appl. 122 (2012) 1397-1436. | MR 2914757 | Zbl 1266.82080

[4] P. M. Bleher. The renormalization group on hierarchical lattices. In Stochastic Methods in Mathematics and Physics (Karpacz, 1988) 171-201. World Sci. Publ., Teaneck, NJ, 1989. | MR 1124692

[5] D. Cheliotis and F. Den Hollander. Variational characterization of the critical curve for pinning of random polymers. Ann. Probab. 41 (2013) 1767-1805. | MR 3098058 | Zbl 1281.60083

[6] P. Collet, J.-P. Eckmann, V. Glaser and A. Martin. Study of the iterations of a mapping associated to a spin glass model. Commun. Math. Phys. 94 (1984) 353-370. | MR 763384 | Zbl 0553.60099

[7] B. Derrida and E. Gardner. Renormalization group study of a disordered model. J. Phys. A 17 (1984) 3223-3236. | MR 772152

[8] B. Derrida, G. Giacomin, H. Lacoin and F. L. Toninelli. Fractional moment bounds and disorder relevance for pinning models. Commun. Math. Phys. 287 (2009) 867-887. | MR 2486665 | Zbl 1226.82028

[9] B. Derrida, V. Hakim and J. Vannimenus. Effect of disorder on two-dimensional wetting. J. Statist. Phys. 66 (1992) 1189-1213. | MR 1156401 | Zbl 0900.82051

[10] F. J. Dyson. Existence of a phase-transition in a one-dimensional Ising ferromagnet. Commun. Math. Phys. 12 (1969) 91-107. | MR 436850

[11] G. Giacomin. Random Polymer Models. Imperial College Press, London, 2007. | MR 2380992 | Zbl 1125.82001

[12] G. Giacomin. Renewal convergence rates and correlation decay for homogeneous pinning models. Electron. J. Probab. 13 (2008) 513-529. | MR 2386741 | Zbl 1190.60086

[13] G. Giacomin. Disorder and Critical Phenomena Through Basic Probability Models. Lecture Notes in Mathematics 2025. Springer, Berlin, 2011. | MR 2816225 | Zbl 1230.82004

[14] G. Giacomin, H. Lacoin and F. Toninelli. Hierarchical pinning model, quadratic maps and quenched disorder. Probab. Theory Related Fields 148 (2010) 159-175. | MR 2653225 | Zbl 1201.60095

[15] G. Giacomin, H. Lacoin and F. L. Toninelli. Marginal relevance of disorder for pinning models. Commun. Pure Appl. Math. 63 (2010) 233-265. | MR 2588461 | Zbl 1189.60173

[16] G. Giacomin, H. Lacoin and F. L. Toninelli. Disorder relevance at marginality and critical point shift. Ann. Inst. Henri Poincaré Probab. Stat. 47 (2011) 148-175. | Numdam | MR 2779401 | Zbl 1210.82036

[17] G. Giacomin and F. L. Toninelli. Smoothing effect of quenched disorder on polymer depinning transitions. Commun. Math. Phys. 266 (2006) 1-16. | MR 2231963 | Zbl 1113.82032

[18] A. B. Harris. Effect of random defects on the critical behaviour of ising models. J. Phys. C 7 (1974) 1671-1692.

[19] H. Lacoin. The martingale approach to disorder irrelevance for pinning models. Electron. Commun. Probab. 15 (2010) 418-427. | MR 2726088 | Zbl 1221.82058

[20] H. Lacoin and F. L. Toninelli. A smoothing inequality for hierarchical pinning models. In Spin Glasses: Statics and Dynamics 271-278. A. Boutet de Monvel and A. Bovier (eds.). Progress in Probability 62. Birkhäuser Verlag, Basel, 2009. | MR 2761990 | Zbl 1194.82043

[21] J. Poisat. On quenched and annealed critical curves of random pinning model with finite range correlations. Ann. Inst. Henri Poincaré Probab. Stat. 49 (2013) 456-482. | Numdam | MR 3088377 | Zbl 1276.82024

[22] F. L. Toninelli. Critical properties and finite-size estimates for the depinning transition of directed random polymers. J. Statist. Phys. 126 (2007) 1025-1044. | MR 2311896 | Zbl 1122.82053

[23] F. L. Toninelli. A replica-coupling approach to disordered pinning models. Commun. Math. Phys. 280 (2008) 389-401. | MR 2395475 | Zbl 1207.82026

[24] A. Weinrib and B. I. Halperin. Critical phenomena in systems with long-range-correlated quenched disorder. Phys. Rev. B 27 (1983) 413-427.