Abstract
We study a hierarchical disordered pinning model with site disorder for which, like in the bond disordered case (Derrida et al. in J Stat Phys 66:1189–1213, 1992 and Giacomin et al. in Probab. Theor. Rel. Fields 2009, arXiv:0711.4649 [math.PR]), there exists a value of a parameter b (which enters in the definition of the hierarchical lattice) that separates an irrelevant disorder regime and a relevant disorder regime. We show that for such a value of b the critical point of the disordered system is different from the critical point of the annealed version of the model. The proof goes beyond the technique used in Giacomin et al. (Probab. Theor. Rel. Fields 2009, arXiv:0711.4649 [math.PR]) and it takes explicitly advantage of the inhomogeneous character of the Green function of the model.
References
Alexander K.S.: The effect of disorder on polymer depinning transitions. Commun. Math. Phys. 279, 117–146 (2008)
Alexander, K., Zygouras, N.: Quenched and annealed critical points in polymer pinning models (2008, preprint) arXiv:0805.1708v1[math.PR]
Alexander, K., Zygouras, N.: Equality of critical points for polymer depinning transitions with loop exponent one (2008, preprint) arXiv:0811.1902v1 [math.PR]
Derrida B., Gardner E.: Renormalisation group study of a disordered model. J. Phys. A Math. Gen. 17, 3223–3236 (1984)
Derrida B., Giacomin G., Lacoin H., Toninelli F.L.: Fractional moment bounds and disorder relevance for pinning models. Commun. Math. Phys. 287, 867–887 (2009)
Derrida B., Hakim V., Vannimenius J.: Effect of disorder on two-dimensional wetting. J. Stat. Phys. 66, 1189–1213 (1992)
Forgacs G., Luck J.M., Nieuwenhuizen Th.M., Orland H.: Wetting of a disordered substrate: exact critical behavior in two dimensions. Phys. Rev. Lett. 57, 2184–2187 (1986)
Giacomin G.: Random Polymer Models. IC press, World Scientific, London (2007)
Giacomin, G., Lacoin, H., Toninelli, F.L.: Hierarchical pinning models, quadratic maps and quenched disorder. Probab. Theor. Rel. Fields (2009, to appear) arXiv:0711.4649 [math.PR]
Giacomin, G., Lacoin, H., Toninelli, F.L.: Marginal relevance of disorder for pinning models (2008, preprint) arXiv:0811.0723 [math-ph]
Giacomin G., Toninelli F.L.: Smoothing effect of quenched disorder on polymer depinning transitions. Commun. Math. Phys. 266, 1–16 (2006)
Toninelli F.L.: A replica-coupling approach to disordered pinning models. Commun. Math. Phys. 280, 389–401 (2008)
Toninelli F.L.: Disordered pinning models and copolymers: beyond annealed bounds. Ann. Appl. Probab. 18, 1569–1587 (2008)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Lacoin, H. Hierarchical pinning model with site disorder: disorder is marginally relevant. Probab. Theory Relat. Fields 148, 159–175 (2010). https://doi.org/10.1007/s00440-009-0226-6
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00440-009-0226-6
Keywords
- Hierarchical pinning models
- Diamond lattices
- Quenched disorder
- Critical behavior
Mathematics Subject Classification (2000)
- 60K37
- 82B44
- 37H99