Asymmetric directed polymers in random environments

Abstract

The model of Brownian Percolation has been introduced as an approximation of discrete last-passage percolation models close to the axis. It allowed to compute some explicit limits and prove fluctuation theorems for these, based on the relations between the Brownian percolation and random matrices. Here, we present two approaches that allow to treat discrete asymmetric models of directed polymers. In both cases, the behaviour is universal, meaning that the results do not depend on the precise law of the environment as long as it satisfies some natural moment assumptions. First, we establish an approximation of asymmetric discrete directed polymers in random environments at very high temperature by a continuous-time directed polymers model in a Brownian environment, much in the same way than the last passage percolation case. The key ingredient is a strong embedding argument developed by Kómlos, Major and Túsnady. Then, we study the partition function of a $1+1$-dimensional directed polymer in a random environment with a drift tending to infinity. We give an explicit expression for the free energy based on known asymptotics for last-passage percolation and compute the order of the fluctuations of the partition function. We conjecture that the law of the properly rescaled fluctuations converges to the GUE Tracy-Widom distribution.