Séminaire MODAL'X : Niccolò Torri (MODAL'X)

A classical problem in the field of disordered systems is to understand the behaviour of a directed polymer in interaction with a random environment. Mathematically, the directed polymer is modeled by a directed random walk and the random environment is a sequence of random variables which can interact with the random walk perturbing its behaviour, giving rise to super-diffusive transversal fluctuations and localisation phenomena. Understanding the typical trajectories of the the walk is a very challenging issue and several results have been obtained when the random environment has exponential moments. In this talk we consider the case of a heavy-tail random environment, that is, the environment’s distribution function decades polynomially with exponent $\alpha >0$. We mainly focus on the case in which the random environment has no second moment ($\alpha\in (0,2)$), finding the super-diffusive transversal fluctuations of the polymer as function of the parameter $\alpha$.

– Joint work with Quentin Berger (Sorbonne Université).