Abstract
We study random pinning and copolymer models, when the return distribution of the underlying renewal process has a polynomial tail with finite mean. We compute the asymptotic behavior of the critical curves of the models in the weak coupling regime, showing that it is universal. This proves a conjecture of Bolthausen, den Hollander and Opoku for copolymer models (Bolthausen et al., in Ann Probab, 2012), which we also extend to pinning models.
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Alexander K.S.: The effect of disorder on polymer depinning transitions. Commun. Math. Phys. 279, 117–146 (2008)
Alexander K.S.: Excursions and local limit theorems for Bessel-like random walks. Electron. J. Prob. 16, 1–44 (2011)
Alexander K.S., Zygouras N.: Quenched and annealed critical points in polymer pinning models. Commun. Math. Phys. 291, 659–689 (2010)
Alexander K.S., Zygouras N.: Equality of critical points for polymer depinning transitions with loop exponent one. Ann. Appl. Prob 20, 356–366 (2010)
Bodineau T., Giacomin G.: On the localization transition of random copolymers near selective interfaces. J. Stat. Phys. 117, 801–818 (2004)
Bodineau T., Giacomin G., Lacoin H., Toninelli F.L.: Copolymers at selective interfaces: new bounds on the phase diagram. J. Stat. Phys. 132, 603–626 (2008)
Bolthausen E., den Hollander F.: Localization transition for a polymer near an interface. Ann. Probab. 25, 1334–1366 (1997)
Bolthausen, E., den Hollander, F., Opoku, A.A.: A copolymer near a selective interface: variational characterization of the free energy. Ann. Probab. (2012, to appear). http://arxiv.org/abs/1110.1315v2 [math.PR]
Caravenna F., den Hollander F.: A general smoothing inequality for disordered polymers. Electron. Commun. Probab. 18(76), 1–15 (2013)
Caravenna F., Giacomin G.: The weak coupling limit of disordered copolymer models. Ann. Probab. 38, 2322–2378 (2010)
Caravenna F., Giacomin G., Gubinelli M.: A numerical approach to copolymers at selective interfaces. J. Stat. Phys. 122, 799–832 (2006)
Caravenna, F., Giacomin, G., Toninelli, F.L.: Copolymers at selective interfaces: settled issues and open problems. In: Probability in Complex Physical Systems. In honour of Erwin Bolthausen and Jürgen Gärtner. Springer Proceedings in Mathematics, Vol. 11, Berlin-Heidelberg-New York: Springer, 2012, pp. 289–311
Caravenna, F., Sun, R., Zygouras, N.: The continuum disordered pinning model. In preparation
Caravenna, F., Sun, R., Zygouras, N.: Polynomial chaos and scaling limits of disordered systems. In preparation
Cheliotis D., den Hollander F.: Variational characterization of the critical curve for pinning of random polymers. Ann. Probab. 41(33), 1767–1805 (2013)
Dembo, A., Zeitouni, O.: Large Deviations Techniques and Applications (2nd. ed.). Berlin-Heidelberg-New York: Springer, 1998
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)
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. London: Imperial College Press, 2007
Giacomin, G.: Disorder and critical phenomena through basic probability models. In: Lecture Notes from the 40th Probability Summer School held in Saint-Flour, 2010, Berlin-Heidelberg-New York: Springer, 2011
Giacomin G., Lacoin H., Toninelli F.L.: Disorder relevance at marginality and critical point shift. Ann. Inst. H. Poincaré Probab. Stat. 47, 148–175 (2011)
Giacomin G., Lacoin H., Toninelli F.L.: Marginal relevance of disorder for pinning models. Commun. Pure Appl. Math. 63, 233–2650 (2011)
Giacomin G., Toninelli F.L.: Smoothing effect of quenched disorder on polymer depinning transitions. Commun. Math. Phys. 266, 1–16 (2006)
den Hollander, F.: Random polymers. In: Lectures from the 37th Probability Summer School held in Saint-Flour 2007. Berlin: Springer-Verlag, 2009
Lacoin H.: The martingale approach to disorder irrelevance for pinning models. Electron. Commun. Probab. 15, 418–427 (2010)
Monthus C.: On the localization of random heteropolymers at the interface between two selective solvents. Eur. Phys. J. B 13, 111–130 (2000)
Nelson D.R., Vinokur V.M.: Boson localization and correlated pinning of superconducting vortex arrays. Phys. Rev. B 48, 13060–13097 (1993)
Poland, D., Scheraga, H.: Theory of helix-coil transitions in biopolymers: statistical mechanical theory of order-disorder transitions in biological macromolecules, London-New York: Academic Press, 1970
Toninelli F.L.: A replica-coupling approach to disordered pinning models. Commun. Math. Phys. 280, 389–401 (2008)
Toninelli F.L.: Coarse graining, fractional moments and the critical slope of random copolymers. Electron. J. Probab. 14, 531–547 (2009)
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Communicated by F. L. Toninelli
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Berger, Q., Caravenna, F., Poisat, J. et al. The Critical Curves of the Random Pinning and Copolymer Models at Weak Coupling. Commun. Math. Phys. 326, 507–530 (2014). https://doi.org/10.1007/s00220-013-1849-0
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DOI: https://doi.org/10.1007/s00220-013-1849-0