We study the probability $p\equiv p_{\eta }\left(t\right)$ 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 $\overline{p^m}\simeq {\gamma }_m/t$. From this, we extract the exact tail $\sim \rho \left(p\right)/t$ of the probability distribution of the noncrossing probability at large time. The exact formula is compared to numerical simulations, with excellent agreement.

• Received 25 November 2015

DOI:https://doi.org/10.1103/PhysRevE.93.032118