The model of directed polymers describe the behavior of a long, directed chain that spreads among an inhomogeneous environment which may attract or repulse the polymer. When the spacial dimension is larger than three, a phase transition occurs between diffusivity (high temperature) and localization (low temperature). On the other hand, in dimensions one and two the polymer is always localized. Dimension two is however critical, as one can recover a phase transition by letting the temperature tend to infinity under a specific parametrization (Caravenna-Sun-Zygouras 17’). In this talk, I will present some of the main results that are known about this scaling regime, and discuss the recent advances that have occurred in the past few years. In particular, I will describe some results that I have obtained with my coauthors (Anna Donadini, Shuta Nakajima and Ofer Zeitouni) on the diffusive phase and its relation to Gaussian logarithmically correlated fields. I will also discuss connexions of the model with the Kardar-Parisi-Zhang (KPZ) equation and the stochastic heat equation.
