The 2d directed polymer under the quasi-critical regime

Cottini Francesca, University of Luxembourg - National University of Singapore

The directed polymer in random environment describes a perturbation of the simple random walk given by a random environment (disorder). The partition function of this model has been widely investigated in recent years, also motivated by its link with the solution of the Stochastic Heat Equation. In dimension 2, the asymptotic behavior of the partition function exhibits a phase transition from the subcritical regime (where a deep understanding has by now been obtained) to the critical regime (where many key questions are still open). The aim of this talk is to introduce a novel regime, named quasi-critical regime, which interpolates between the other two and is arbitrarily “close” to the critical one. We show that this is the most extended regime where Gaussian fluctuations can hold, before reaching the critical regime where they fail. If time allows, we will also discuss the large-scale behavior of novel local averages of the partition function on sub-diffusive scales under the quasi-critical regime.

Area: IS16 - Random walks and disordered models (Niccolò Torri)

Keywords: Directed polymer, partition functions, polynomial chaos, Stochastic Heat Equation, disordered systems

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