Crossing probability for directed polymers in random media. II. Exact tail of the distribution

We study the probability p = p(eta)(t) that two directed polymers in a given random potential eta and with fixed and nearby endpoints do not cross until time t. This probability is itself a random variable (over samples eta), which, as we show, acquires a very broad probability distribution at large time. In particular, the moments of p are found to be dominated by atypical samples where p is of order unity. Building on a formula established by us in a previous work using nested Bethe ansatz and Macdonald process methods, we obtain analytically the leading large time behavior of all moments (p(m)) over bar similar or equal to gamma(m)/t. From this, we extract the exact tail similar to rho(p)/t of the probability distribution of the noncrossing probability at large time. The exact formula is compared to numerical simulations, with excellent agreement.