Free energy fluctuations for directed polymers in 1+1 dimension

The Kardar-Parisi-Zhang (KPZ) universality class includes directed polymers in random media in 1+1 dimension. According to the universality conjecture, for any finite temperature, the fluctuations of the free energy (e.g. for point-to-point) directed polymers is expected to be distributed as the (GUE) Tracy-Widom distribution in the limit of large system size. This distribution arose first in the context of random matrices.

Detailed results as the fluctuation laws for models in the KPZ were, until recently, available only for "zero temperature models". We consider two models at positive temperature, a semi-discrete and the continuum directed polymer models, and determine the law of the free energy fluctuations. Results for the stationary KPZ solution will be also mentioned.

This talk is based on a joint work with Alexei Borodin and Ivan Corwin <link http: arxiv.org abs external>arxiv.org/abs/1204.10242, their previous work <link http: arxiv.org abs external>arxiv.org/abs/1111.4408, as well as a work in progress with them and Balint Veto.