We consider d-dimensional random surface models which for d=1 are the standard (tied-down) random walks (considered as a random ``string''). In higher dimensions, the one-dimensional (discrete) time parameter of the random walk is replaced by the d-dimensional lattice \Z^d, or a finite subset of it. The random surface is represented by real-valued random variables \phi_i, where i is in \Z^d. A class of natural generalizations of the standard random walk are gradient models whose laws are (formally) expressed as

P(d\phi) = 1/Z \exp[-\sum_{|i-j|=1}V(\phi_i-\phi_j)] \prod_i d\phi_i,

V:\R -> R^+ convex, and with some growth conditions. Such surfaces have been

introduced in theoretical physics as (simplified) models for random interfaces separating different phases. Of particular interest are localization-delocalization phenomena, for instance for a surface interacting with a wall by attracting or repulsive interactions, or both together. Another example are so-called heteropolymers which have a noise-induced interaction. Recently, there had been developments of new probabilistic tools for such problems. Among them are: o Random walk representations of Helffer-Sjöstrand type, o Multiscale analysis, o Connections with random trapping problems and large deviations We give a survey of some of these developments.

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Bolthausen, E (2002). *Localization-delocalization phenomena for random interfaces.* In: Tatsien, L. Proceedings of the International Congress of Mathematicians, Vol. III (Beijing, 2002). Beijing: Higher Education Press, 25-39.

## Abstract

We consider d-dimensional random surface models which for d=1 are the standard (tied-down) random walks (considered as a random ``string''). In higher dimensions, the one-dimensional (discrete) time parameter of the random walk is replaced by the d-dimensional lattice \Z^d, or a finite subset of it. The random surface is represented by real-valued random variables \phi_i, where i is in \Z^d. A class of natural generalizations of the standard random walk are gradient models whose laws are (formally) expressed as

P(d\phi) = 1/Z \exp[-\sum_{|i-j|=1}V(\phi_i-\phi_j)] \prod_i d\phi_i,

V:\R -> R^+ convex, and with some growth conditions. Such surfaces have been

introduced in theoretical physics as (simplified) models for random interfaces separating different phases. Of particular interest are localization-delocalization phenomena, for instance for a surface interacting with a wall by attracting or repulsive interactions, or both together. Another example are so-called heteropolymers which have a noise-induced interaction. Recently, there had been developments of new probabilistic tools for such problems. Among them are: o Random walk representations of Helffer-Sjöstrand type, o Multiscale analysis, o Connections with random trapping problems and large deviations We give a survey of some of these developments.

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## Additional indexing

Item Type: | Book Section, refereed, original work |
---|---|

Communities & Collections: | 07 Faculty of Science > Institute of Mathematics |

Dewey Decimal Classification: | 510 Mathematics |

Uncontrolled Keywords: | random string; lattice; gradient model |

Language: | English |

Date: | 2002 |

Deposited On: | 27 Apr 2010 06:34 |

Last Modified: | 05 Apr 2016 13:25 |

Publisher: | Higher Education Press |

ISBN: | 7-04-008690-5 |

Official URL: | http://www.hep.edu.cn/cooperate/order/4.htm |

Related URLs: | http://www.ams.org/mathscinet-getitem?mr=1957516 http://www.zentralblatt-math.org/zmath/en/search/?q=an:1006.60099 |

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