  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 K\'omlos, Major and T\'usnady.
  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.
