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You are here: Home en Opencourse SGU Mathematics Kickoff Meeting video Random polymers

Random polymers

A (directed) random polymer in d + 1 dimensions is a (standard) random walk in d dimensions where the time axis is considered as an additional dimension, which is transformed by a random potential in space and time. The best known and most famous case is the directed polymer in random environment which has a potential given by independent random variables in space and time. Some of the basic questions are open even in 1 + 1 dimensions which is believed to be connected with the KPZ universality class. The only case of the 1 + 1 dimensional directed polymer which has been fully analyzed is a very special one investigated by Kurt Johannson. Important partial results on the directed polymers have been obtained by Imbrie-Spencer, Bolthausen, Comets-Shiga-Yoshida, Carmona, Petermann, and many others.
The main focus of the talk will however be on the so-called copolymer, first discussed in the physics literature by Garel, Huse, Leibler and Orland in 1989 which models the behavior of a polymer at an interface. Important rigorous results have first been obtained by Sinai and Bolthausen-den Hollander, and have later further been developed by Bodineau, Giacomin, Caravenna and others. A basic object of interest is a critical line in the parameter space which separates a localized phase from a delocalized one. Particularly interesting is the behavior at the weak disorder limit where the phase transition is characterized by a universal critical tangent whose existence had first been proved in the Bolthausen-den Hollander paper, and whose exact value is still open. In a recent paper by Bolthausen-den Hollander-Opoku, a new lower bound has been derived, disproving a long-standing conjecture from the physics literature. The bound was obtained by an application of a sophisticated large deviation theory developed by Birkner-Greven-den Hollander. We present a sketch of an elementary version of this new bound.