Mathematics – Probability
Scientific paper
2003-04-24
Proceedings of the ICM, Beijing 2002, vol. 3, 25--40
Mathematics
Probability
Scientific paper
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-Sj\"ostrand 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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