The search session has expired. Please query the service again.

The search session has expired. Please query the service again.

The search session has expired. Please query the service again.

The search session has expired. Please query the service again.

The search session has expired. Please query the service again.

The search session has expired. Please query the service again.

The search session has expired. Please query the service again.

The search session has expired. Please query the service again.

The search session has expired. Please query the service again.

Hierarchical pinning model in correlated random environment

Quentin Berger; Fabio Lucio Toninelli

Annales de l'I.H.P. Probabilités et statistiques (2013)

  • Volume: 49, Issue: 3, page 781-816
  • ISSN: 0246-0203

Abstract

top
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 Fields148 (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.

How to cite

top

Berger, Quentin, and Toninelli, Fabio Lucio. "Hierarchical pinning model in correlated random environment." Annales de l'I.H.P. Probabilités et statistiques 49.3 (2013): 781-816. <http://eudml.org/doc/272100>.

@article{Berger2013,
abstract = {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 Fields148 (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.},
author = {Berger, Quentin, Toninelli, Fabio Lucio},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {pinning models; polymer; disordered models; Harris criterion; critical phenomena; correlation; hierarchical models},
language = {eng},
number = {3},
pages = {781-816},
publisher = {Gauthier-Villars},
title = {Hierarchical pinning model in correlated random environment},
url = {http://eudml.org/doc/272100},
volume = {49},
year = {2013},
}

TY - JOUR
AU - Berger, Quentin
AU - Toninelli, Fabio Lucio
TI - Hierarchical pinning model in correlated random environment
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2013
PB - Gauthier-Villars
VL - 49
IS - 3
SP - 781
EP - 816
AB - 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 Fields148 (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.
LA - eng
KW - pinning models; polymer; disordered models; Harris criterion; critical phenomena; correlation; hierarchical models
UR - http://eudml.org/doc/272100
ER -

References

top
  1. [1] K. S. Alexander. The effect of disorder on polymer depinning transitions. Commun. Math. Phys.279 (2008) 117–146. Zbl1175.82034MR2377630
  2. [2] K. S. Alexander and N. Zygouras. Quenched and annealed critical points in polymer pinning models. Commun. Math. Phys.291 (2009) 659–689. Zbl1188.82154MR2534789
  3. [3] Q. Berger and H. Lacoin. Sharp critical behavior for random pinning model with correlated environment. Stochastic Process. Appl.122 (2012) 1397–1436. Zbl1266.82080MR2914757
  4. [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. MR1124692
  5. [5] D. Cheliotis and F. den Hollander. Variational characterization of the critical curve for pinning of random polymers. Ann. Probab.41 (2013) 1767–1805. Zbl1281.60083MR3098058
  6. [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. Zbl0553.60099MR763384
  7. [7] B. Derrida and E. Gardner. Renormalization group study of a disordered model. J. Phys. A17 (1984) 3223–3236. MR772152
  8. [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. Zbl1226.82028MR2486665
  9. [9] B. Derrida, V. Hakim and J. Vannimenus. Effect of disorder on two-dimensional wetting. J. Statist. Phys.66 (1992) 1189–1213. Zbl0900.82051MR1156401
  10. [10] F. J. Dyson. Existence of a phase-transition in a one-dimensional Ising ferromagnet. Commun. Math. Phys.12 (1969) 91–107. Zbl1306.47082MR436850
  11. [11] G. Giacomin. Random Polymer Models. Imperial College Press, London, 2007. Zbl1125.82001MR2380992
  12. [12] G. Giacomin. Renewal convergence rates and correlation decay for homogeneous pinning models. Electron. J. Probab.13 (2008) 513–529. Zbl1190.60086MR2386741
  13. [13] G. Giacomin. Disorder and Critical Phenomena Through Basic Probability Models. Lecture Notes in Mathematics 2025. Springer, Berlin, 2011. Zbl1230.82004MR2816225
  14. [14] G. Giacomin, H. Lacoin and F. Toninelli. Hierarchical pinning model, quadratic maps and quenched disorder. Probab. Theory Related Fields148 (2010) 159–175. Zbl1201.60095MR2653225
  15. [15] G. Giacomin, H. Lacoin and F. L. Toninelli. Marginal relevance of disorder for pinning models. Commun. Pure Appl. Math.63 (2010) 233–265. Zbl1189.60173MR2588461
  16. [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. Zbl1210.82036MR2779401
  17. [17] G. Giacomin and F. L. Toninelli. Smoothing effect of quenched disorder on polymer depinning transitions. Commun. Math. Phys.266 (2006) 1–16. Zbl1113.82032MR2231963
  18. [18] A. B. Harris. Effect of random defects on the critical behaviour of ising models. J. Phys. C7 (1974) 1671–1692. 
  19. [19] H. Lacoin. The martingale approach to disorder irrelevance for pinning models. Electron. Commun. Probab.15 (2010) 418–427. Zbl1221.82058MR2726088
  20. [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. Zbl1194.82043MR2761990
  21. [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. Zbl1276.82024MR3088377
  22. [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. Zbl1122.82053MR2311896
  23. [23] F. L. Toninelli. A replica-coupling approach to disordered pinning models. Commun. Math. Phys.280 (2008) 389–401. Zbl1207.82026MR2395475
  24. [24] A. Weinrib and B. I. Halperin. Critical phenomena in systems with long-range-correlated quenched disorder. Phys. Rev. B27 (1983) 413–427. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.