##
**Smoothing effect of quenched disorder on polymer depinning transitions.**
*(English)*
Zbl 1113.82032

Quenched disorder is expected to smooth phase transitions in many situations. In this work under some conditions on the disorder distribution the authors prove that such effect takes place in models of directed polymers in random media exhibiting a localization/delocalization transition on a defect line. Such a transition may be of first or higher order in the corresponding pure non-disordered cases. It is shown that as soon as disorder is present, the transition is at least of second order. Here the mechanism inducing the smoothing of the transition is based on an estimate of the probability that the polymer visits rare but very favorable regions where the disorder produces a large positive fluctuation of the partition function. Two classes of models are considered: random pinning (or wetting) models and random copolymer at selective interfaces.

Reviewer: Boris V. Loginov (Ul’yanovsk)

### MSC:

82B44 | Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics |

82B27 | Critical phenomena in equilibrium statistical mechanics |

82D60 | Statistical mechanics of polymers |

### Keywords:

phase transitions; quenched disorder; polymer depinning transitions; random wetting; directed polymers in random media; random pinning; random copolymer
PDF
BibTeX
XML
Cite

\textit{G. Giacomin} and \textit{F. L. Toninelli}, Commun. Math. Phys. 266, No. 1, 1--16 (2006; Zbl 1113.82032)

### References:

[1] | Aizenman M., Wehr J. (1990) Rounding effects of quenched randomness on first–order phase transitions. Community Math. Phys. 130, 489–528 · Zbl 0714.60090 |

[2] | Alexander, K. S., Sidoravicius, V. Pinning of polymers and interfaces by random potentials. preprint (2005). http://arxiv.org/list/math.PR/0501028, 2005 · Zbl 1145.82010 |

[3] | Bingham N.H., Goldie C.M., Teugels J.L. (1987) Regular Variation. Cambridge University Press, Cambridge · Zbl 0617.26001 |

[4] | Bodineau T., Giacomin G. (2004) On the localization transition of random copolymers near selective interfaces. J. Stat. Phys. 117, 801–818 · Zbl 1089.82031 |

[5] | Bolthausen E., den Hollander F. (1997) Localization transition for a polymer near an interface. Ann. Probab. 25, 1334–1366 · Zbl 0885.60022 |

[6] | Bovier A., Külske C. (1996) There are no nice interfaces in (2+1)–dimensional SOS models in random media. J. Stat. Phys. 83, 751–759 · Zbl 1081.82571 |

[7] | Chayes J.T., Chayes L., Fisher D.S., Spencer T. (1989) Correlation Length Bounds for Disordered Ising Ferromagnets. Commun. Math. Phys. 120, 501–523 · Zbl 0658.60137 |

[8] | Coluzzi B. (2006) Numerical study on a disordered model for DNA denaturation transition. Phys. Rev. E. 73, 011911 |

[9] | Cule D., Hwa T. (1997) Denaturation of heterogeneous DNA. Phys. Rev. Lett. 79, 2375–2378 |

[10] | Derrida B., Hakim V., Vannimenius J. (1992) Effect of disorder on two–dimensional wetting. J. Stat. Phys. 66, 1189–1213 · Zbl 0900.82051 |

[11] | Feller W. (1968) An introduction to probability theory and its applications. Vol. I, Third edition, John Wiley & Sons, Inc., New York–London–Sydney · Zbl 0155.23101 |

[12] | Feller W. (1971) An introduction to probability theory and its applications. Vol. II, Second edition, John Wiley & Sons, Inc., New York–London–Sydney · Zbl 0219.60003 |

[13] | Forgacs G., Luck J.M.Th., Nieuwenhuizen M., Orland H. (1986) Wetting of a Disordered Substrate: Exact Critical behavior in Two Dimensions. Phys. Rev. Lett. 57, 2184–2187 |

[14] | Garel T., Huse D.A., Leibler S., Orland H. (1989) Localization transition of random chains at interfaces. Europhys. Lett. 8, 9–13 |

[15] | Garel, T., Monthus, C. Numerical study of the disordered Poland–Scheraga model of DNA denaturation. J. Stat. Mech., Theory and Experiments (2005), P06004 |

[16] | Giacomin, G. Localization phenomena in random polymer models. Preprint, 2004; Available online: http://www.proba.jussieu.fr/pageperso/giacomin/pub/publicat.html, 2004 |

[17] | Giacomin G., Toninelli F.L. (2005) Estimates on path delocalization for copolymers at selective interfaces. Probab. Theor. Rel. Fields 133, 464–482 · Zbl 1098.60089 |

[18] | Giacomin G., Toninelli F.L. (2006) The localized phase of disordered copolymers with adsorption. ALEA 1, 149–180 · Zbl 1134.82006 |

[19] | Harris A.B. (1974) Effect of random defects on the critical behaviour of Ising models. J. Phys. C 7, 1671–1692 |

[20] | Imry Y., Ma S.–K. (1975) Random–Field Instability of the Ordered State of Continuous Symmetry. Phys. Rev. Lett. 35, 1399–1401 |

[21] | Kafri Y., Mukamel D., Peliti L. (2000) Why is the DNA denaturation transition first order. Phys. Rev. Lett. 85, 4988–4991 |

[22] | Kingman J.F.C. (1973) Subadditive ergodic theory. Ann. Probab. 1, 882–909 · Zbl 0311.60018 |

[23] | Monthus C. (2000) On the localization of random heteropolymers at the interface between two selective. Eur. Phys. J. B 13, 111–130 |

[24] | Petrelis, N. Polymer pinning at an interface. Preprint, 2005; available on: http://arxiv.org/list/math.PR/0504464, 2005 |

[25] | Sinai G., Ya. (1993) A random walk with a random potential. Theory Probab. Appl. 38, 382–385 · Zbl 04520827 |

[26] | Soteros C.E., Whittington S.G. (2004) The statistical mechanics of random copolymers. J. Phys. A: Math. Gen. 37, R279–R325 · Zbl 1073.82015 |

[27] | Tang L.–H., Chaté H. (2001) Rare–Event Induced Binding Transition of Heteropolymers. Phys. Rev. Lett. 86, 830–833 |

[28] | Trovato T., Maritan A. (1999) A variational approach to the localization transition of heteropolymers at interfaces. Europhys. Lett. 46, 301–306 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.