The Annals of Applied Probability

Disordered pinning models and copolymers: Beyond annealed bounds

Fabio Lucio Toninelli

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Abstract

We consider a general model of a disordered copolymer with adsorption. This includes, as particular cases, a generalization of the copolymer at a selective interface introduced by Garel et al. [Europhys. Lett. 8 (1989) 9–13], pinning and wetting models in various dimensions, and the Poland–Scheraga model of DNA denaturation. We prove a new variational upper bound for the free energy via an estimation of noninteger moments of the partition function. As an application, we show that for strong disorder the quenched critical point differs from the annealed one, for example, if the disorder distribution is Gaussian. In particular, for pinning models with loop exponent 0<α<1/2 this implies the existence of a transition from weak to strong disorder. For the copolymer model, under a (restrictive) condition on the law of the underlying renewal, we show that the critical point coincides with the one predicted via renormalization group arguments in the theoretical physics literature. A stronger result holds for a “reduced wetting model” introduced by Bodineau and Giacomin [J. Statist. Phys. 117 (2004) 801–818]: without restrictions on the law of the underlying renewal, the critical point coincides with the corresponding renormalization group prediction.

Article information

Source
Ann. Appl. Probab., Volume 18, Number 4 (2008), 1569-1587.

Dates
First available in Project Euclid: 21 July 2008

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1216677132

Digital Object Identifier
doi:10.1214/07-AAP496

Mathematical Reviews number (MathSciNet)
MR2434181

Zentralblatt MATH identifier
1157.60090

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.) 60K05: Renewal theory

Keywords
Pinning and wetting models copolymers at selective interfaces annealed bounds fractional moments

Citation

Toninelli, Fabio Lucio. Disordered pinning models and copolymers: Beyond annealed bounds. Ann. Appl. Probab. 18 (2008), no. 4, 1569--1587. doi:10.1214/07-AAP496. https://projecteuclid.org/euclid.aoap/1216677132


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References

  • [1] Aizenman, M. and Molchanov, S. (1993). Localization at large disorder and at extreme energies: An elementary derivation. Comm. Math. Phys. 157 245–278.
  • [2] Alexander, K. S. (2008). The effect of disorder on polymer depinning transitions. Comm. Math. Phys. 279 117–146.
  • [3] Alexander, K. S. and Sidoravicius, V. (2006). Pinning of polymers and interfaces by random potentials. Ann. Appl. Probab. 16 636–669.
  • [4] Biskup, M. and den Hollander, F. (1999). A heteropolymer near a linear interface. Ann. Appl. Probab. 9 668–687.
  • [5] Bodineau, T. and Giacomin, G. (2004). On the localization transition of random copolymers near selective interfaces. J. Statist. Phys. 117 801–818.
  • [6] Bolthausen, E., Caravenna, F. and de Tilière, B. (2007). The quenched critical point of a diluted disordered polymer model. Preprint. ArXiv:math.0711.0141v1 [math.PR].
  • [7] Bolthausen, E. and den Hollander, F. (1997). Localization transition for a polymer near an interface. Ann. Probab. 25 1334–1366.
  • [8] Buffet, E., Patrick, A. and Pulé, J. V. (1993). Directed polymers on trees: A martingale approach. J. Phys. A 26 1823–1834.
  • [9] Caravenna, F., Giacomin, G. and Gubinelli, M. (2006). A numerical approach to copolymers at selective interfaces. J. Statist. Phys. 122 799–832.
  • [10] Caravenna, F. and Giacomin, G. (2005). On constrained annealed bounds for pinning and wetting models. Electron. Comm. Probab. 10 179–189.
  • [11] Cule, D. and Hwa, T. (1997). Denaturation of heterogeneous DNA. Phys. Rev. Lett. 79 2375–2378.
  • [12] Derrida, B., Hakim, V. and Vannimenius, J. (1992). Effect of disorder on two-dimensional wetting. J. Statist. Phys. 66 1189–1213.
  • [13] Evans, M. R. and Derrida, B. (1992). Improved bounds for the transition temperature of directed polymers in a finite-dimensional random medium. J. Statist. Phys. 69 427–437.
  • [14] Forgacs, G., Luck, J. M., Nieuwenhuizen, Th. M. and Orland, H. (1986). Wetting of a disordered substrate: Exact critical behavior in two dimensions. Phys. Rev. Lett. 57 2184–2187.
  • [15] Gangardt, D. M. and Nechaev, S. K. (2007). Wetting transition on a one-dimensional disorder. Preprint. ArXiv:math.0704.2893.
  • [16] Garel, T., Huse, D. A., Leibler, S. and Orland, H. (1989). Localization transition of random chains at interfaces. Europhys. Lett. 8 9–13.
  • [17] Giacomin, G. (2007). Random Polymer Models. Imperial College Press, World Scientific.
  • [18] Giacomin, G. and Toninelli, F. L. (2006). Smoothing effect of quenched disorder on polymer depinning transitions. Comm. Math. Phys. 266 1–16.
  • [19] Giacomin, G. and Toninelli, F. L. (2005). Estimates on path delocalization for copolymers at selective interfaces. Probab. Theory Related Fields 133 464–482.
  • [20] Giacomin, G. and Toninelli, F. L. (2006). The localized phase of disordered copolymers with adsorption. ALEA 1 149–180.
  • [21] Giacomin, G. and Toninelli, F. L. (2007). On the irrelevant disorder regime of pinning models. Preprint. ArXiv:math.0707.3340v1 [math.PR].
  • [22] Hardy, G. H., Littlewood, J. E. and Pólya, G. (1967). Inequalities, 2nd ed. Cambridge Univ. Press.
  • [23] Harris, A. B. (1974). Effect of random defects on the critical behaviour of Ising models. J. Phys. C 7 1671–1692.
  • [24] Kafri, Y., Mukamel, D. and Peliti, L. (2000). Why is the DNA denaturation transition first order? Phys. Rev. Lett. 85 4988–4991.
  • [25] Monthus, C. (2000). On the localization of random heteropolymers at the interface between two selective solvents. Eur. Phys. J. B 13 111–130.
  • [26] Morita, T. (1966). Statistical mechanics of quenched solid solutions with application to magnetically dilute alloys. J. Math. Phys. 5 1401–1405.
  • [27] Nelson, D. R. and Vinokur, V. M. (1993). Boson localization and correlated pinning of superconducting vortex arrays. Phys. Rev. B 48 13060–13097.
  • [28] Toninelli, F. L. (2007). Critical properties and finite-size estimates for the depinning transition of directed random polymers. J. Statist. Phys. 126 1025–1044.
  • [29] Toninelli, F. L. (2008). A replica-coupling approach to disordered pinning models. Comm. Math. Phys. To appear.