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

Andrea De Luca 1 Pierre Le Doussal 2

Physical Review E : Statistical, Nonlinear, and Soft Matter Physics, American Physical Society, 2016, 93, pp.032118. <10.1103/PhysRevE.93.032118>

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.

  • 1. LPTMS – Laboratoire de Physique Théorique et Modèles Statistiques
  • 2. LPTENS – Laboratoire de Physique Théorique de l’ENS
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