Random polymers and random walks in random environments

Abstract

In this project we would like to study two known models from probability. The first is the directed polymer model, which, although it has been studied for several decades, still presents many open problems that are of interest to the scientific community. In particular, a long-standing conjecture holds that the model's Lyapunov exponent being negative (called a very strong disorder regime) implies strong disorder. However, in previous work we have provided a counterexample of the conjecture for random walks where the tail of the distribution is sufficiently heavy. This suggests that the conjecture is not universal, and delving into this fact is very important for a better understanding of the model. In this project we aim to understand the differences between the two notions of strong disorder and a reasonable starting point for this project would be to identify a necessary and sufficient condition for the existence of a strong disorder phase. On the other hand, we would also like to study the model of random walks in dynamic random media. Regarding this second model, the interest is in proving the existence of a well-defined limiting velocity for a particle subject to the influence of a dynamic random environment (such as the symmetrical simple exclusion process) and in characterizing its fluctuations around the mean position. In this context, the random walk goes beyond the particles of the environment, allowing a renewal structure to be established. As a consequence, the behavior of the random walk in these regimes is characterized by Gaussian fluctuations and CLT. However, it is not clear whether this diffusive behavior or even this speed is present for other types of dynamic random environments such as the Totally Asymmetrical Exclusion Process (TASAP) since it is not clear that the techniques used in other random environments (renormalization, sprinkling) would also work in this context. (AU)