The model of directed polymer in a random environment is a fundamental model of interaction between a simple random walk and ambient disorder. This interaction gives rise to complex phenomena and transitions from a central limit theorem to novel statistical behaviors. In the first half of the talk, I will introduce this model, explain some of these novel statistical behaviors, highlight some central questions and recent advances in this field.
In the second half of the talk, I will discuss the half-space directed polymer which is a variant of the directed polymer that studies how polymers behave in the presence of an attractive wall. I will explain how this variant introduces a non-trivial phase transition and discuss what we know mathematically about this variant.
