Crossing probability for directed polymers in random media: exact tail of the distribution

Understand

We study the probability $p \equiv p_\eta(t)$ 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 {\it all moments} $\overline{p^m}\simeq \gamma_m/t$.