I will review recent progress in joint work with Hubert Lacoin (IMPA) about the regularity of the phase transition of the directed polymer model in spatial dimensions $d\ge 3$.

 

This model describes a random path (called polymer in this context) affected by a random environment of i.i.d. weights and has received much attention recently because of its (partly conjectural) relationship to the KPZ universality class. It is known that in dimensions $d\ge 3$, the model undergoes a phase transition from a weak disorder, high temperature phase (where the polymer converge to Brownian motion under the usual scaling) to a strong disorder, low temperature phase where the polymer localize into small, favorable regions and are expected to scale superdiffusively. This phase transition is characterized through the positivity (or not) of an associated martingale.

 

In a number of works, we have analyzed the behavior of this model close to its critical point. We have shown that beyond the critical value ${\beta }_c$ for the disorder intensity beta, the associated martingale converges to zero exponentially fast at a rate $f(\beta )$. The main result of this talk is that we obtain upper and lower bounds on $f({\beta }_c+u)$ as $u\to 0$.

 

Organizer

 

Wai-Kit Lam (NTU)