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

TL;DR

This paper derives the exact large-time tail distribution of the probability that two directed polymers in a random medium do not cross, revealing a broad distribution dominated by atypical samples, supported by analytical and numerical results.

Contribution

We analytically determine the exact tail distribution of the non-crossing probability for directed polymers, extending previous formulas using Bethe Ansatz and Macdonald processes.

Findings

The distribution of non-crossing probability is broad and dominated by atypical samples.

All moments of the non-crossing probability decay as 1/t at large times.

Analytical results show excellent agreement with numerical simulations.

Abstract

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} $\overline{p^m}\simeq \gamma_m/t$. From this, we extract the exact tail $\sim \rho(p)/t$ of the probability distribution of the non-crossing probability at large time. The exact formula is compared to numerical simulations, with excellent agreement.