  We study the probability that two directed polymers in the same random
potential do not intersect. We use the replica method to map the problem onto
the attractive Lieb-Liniger model with generalized statistics between
particles. We obtain analytical expressions for the first few moments of this
probability, and compare them to a numerical simulation of a discrete model at
high-temperature. From these observations, several large time properties of the
non-crossing probabilities are conjectured. Extensions of our formalism to more
general observables are discussed.
