Physics

Mathematics

Stochastic process

Statistical physics

Statistics

Mathematical proof

Phase space

Artificial intelligence

Geometry

Computer science

Nuclear magnetic resonance

Probabilistic logic

Quantum mechanics

Polymer

Impurity

Statistical mechanics

Advanced studies in pure mathematics

Directed polymers in random environment can be thought of as a model of statistical mechanics in which paths of stochastic processes interact with a quenched disorder (impurities), depending on both time and space. We review here main results which have been obtained during the last fifteen years, with proofs to most of the results. The material covers the diffusive behavior of the polymers in weak disorder phase studied by J. Imbrie, T. Spencer, E. Bolthausen, R. Song and X. Y. Zhou [11, 3, 25], and localization of the paths in strong disordered phase recently obtained by P. Carmona, Y. Hu, and the authors of the present article [4, 5].

Probabilistic Analysis of Directed Polymers in a Random Environment: a Review