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

We study the probability $p \equiv p_η(t)$ that two directed polymers in a given random potential $η$ and with fixed and nearby endpoints, do not cross until time $t$. This probability is itself a random variable (over samples $η$) 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} $\overline{p^m}\simeq γ_m/t$. From this, we extract the exact tail $\sim ρ(p)/t$ of the probability distribution of the non-crossing probability at large time. The exact formula is compared to numerical simulations, with excellent agreement.

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