Abstract
We investigate the phase diagram of disordered copolymers at the interface between two selective solvents, and in particular its weak-coupling behavior, encoded in the slope m c of the critical line at the origin. We focus on the directed walk case, which has turned out to be, in spite of the apparent simplicity, extremely challenging. In mathematical terms, the partition function of such a model does not depend on all the details of the Markov chain that models the polymer, but only on the time elapsed between successive returns to zero and on whether the walk is in the upper or lower half plane between such returns. This observation leads to a natural generalization of the model, in terms of arbitrary laws of return times: the most interesting case being the one of return times with power law tails (with exponent 1+α, α=1/2 in the case of the symmetric random walk). The main results we present here are:
-
(1)
the improvement of the known result 1/(1+α)≤m c ≤1, as soon as α>1 for what concerns the upper bound, and down to α≈0.65 for the lower bound.
-
(2)
a proof of the fact that the critical curve lies strictly below the critical curve of the annealed model for every non-zero value of the coupling parameter.
We also provide an argument that rigorously shows the strong dependence of the phase diagram on the details of the return probability (and not only on the tail behavior). Lower bounds are obtained by exhibiting a new localization strategy, while upper bounds are based on estimates of non-integer moments of the partition function.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Albeverio, S., Zhou, X.Y.: Free energy and some sample path properties of a random walk with random potential. J. Stat. Phys. 83, 573–622 (1996)
Biskup, M., den Hollander, F.: A heteropolymer near a linear interface. Ann. Appl. Probab. 9, 668–687 (1999)
Bodineau, T., Giacomin, G.: On the localization transition of random copolymers near selective interfaces. J. Stat. Phys. 117, 801–818 (2004)
Bolthausen, E., Caravenna, F., de Tilière, B.: The quenched critical point of a diluted disordered polymer model. arXiv:0711.0141 [math.PR]
Bolthausen, E., den Hollander, F.: Localization transition for a polymer near an interface. Ann. Probab. 25, 1334–1366 (1997)
Bolthausen, E., den Hollander, F.: Private communication
Caravenna, F., Giacomin, G.: On constrained annealed bounds for pinning and wetting models. Electron. Commun. Probab. 10, 179–189 (2005)
Caravenna, F., Giacomin, G., Gubinelli, M.: A numerical approach to copolymers at selective interfaces. J. Stat. Phys. 122, 799–832 (2006)
Causo, M.S., Whittington, S.G.: A Monte Carlo investigation of the localization transition in random copolymers at an interface. J. Phys. A Math. Gen. 36, L189–L195 (2003)
Den Hollander, F., Pétrélis, N.: On the localized phase of a copolymer in an emulsion: supercritical percolation regime. Preprint, arXiv:0709:1659 (2007)
Derrida, B., Giacomin, G., Lacoin, H., Toninelli, F.L.: Fractional moment bounds and disorder relevance for pinning models. arXiv:0712.2515 [math.PR]
Doney, R.A.: One-sided local large deviations and renewal theorems in the case of infinite mean. Probab. Theory Relat. Fields 107, 451–465 (1997)
Feller, W.: An Introduction to Probability Theory and its Applications, vol. 2, 2nd edn. Wiley, New York (1971)
Garel, T., Huse, D.A., Leibler, S., Orland, H.: Localization transition of random chains at interfaces. Europhys. Lett. 8, 9–13 (1989)
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. arXiv:0711.4649 [math.PR]
Giacomin, G., Toninelli, F.L.: Estimates on path delocalization for copolymers at selective interfaces. Probab. Theory Relat. Fields 133, 464–482 (2005)
Giacomin, G., Toninelli, F.L.: The localized phase of disordered copolymers with adsorption. ALEA 1, 149–180 (2006)
Giacomin, G., Toninelli, F.L.: Smoothing effect of quenched disorder on polymer depinning transitions. Commun. Math. Phys. 266, 1–16 (2006)
Habibzadah, N., Iliev, G.K., Martin, R., Saguia, A., Whittington, S.G.: The order of the localization transition for a random copolymer. J. Phys. A Math. Gen. 39, 5659–5667 (2006)
Iliev, G., Rechnitzer, A., Whittington, S.G.: Localization of random copolymers and the Morita approximation. J. Phys. A Math. Gen. 38, 1209–1223 (2005)
Monthus, C.: On the localization of random heteropolymers at the interface between two selective solvents. Eur. Phys. J. B 13, 111–130 (2000)
Monthus, C., Garel, T.: Delocalization transition of the selective interface model: distribution of pseudo-critical temperatures, J. Stat. Mech., P12011 (2005)
Sinai, Ya.G.: A random walk with a random potential. Theory Probab. Appl. 38, 382–385 (1993)
Soteros, C.E., Whittington, S.G.: The statistical mechanics of random copolymers. J. Phys. A Math. Gen. 37, R279–R325 (2004)
Stepanow, S., Sommer, J.-U., Ya, I.: Erukhimovich, Localization transition of random copolymers at interfaces. Phys. Rev. Lett. 81, 4412–4416 (1998)
Toninelli, F.L.: Disordered pinning models and copolymers: beyond annealed bounds. Ann. Appl. Probab. (to appear). arXiv:0709.1629v1 [math.PR]
Trovato, A., Maritan, A.: A variational approach to the localization transition of heteropolymers at interfaces. Europhys. Lett. 46, 301–306 (1999)
Whittington, S.G.: Randomly coloured self-avoiding walks: adsorption and localization. Markov Process. Relat. Fields 13, 761–776 (2007)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bodineau, T., Giacomin, G., Lacoin, H. et al. Copolymers at Selective Interfaces: New Bounds on the Phase Diagram. J Stat Phys 132, 603–626 (2008). https://doi.org/10.1007/s10955-008-9579-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-008-9579-y