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

Andrea De Luca1,* and Pierre Le Doussal2,†

  • 1LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
  • 2Laboratoire de Physique Théorique de l'ENS, CNRS and Ecole Normale Supérieure de Paris, Paris, France

  • *andrea.deluca@lptms.u-psud.fr
  • †ledou@lpt.ens.fr
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Phys. Rev. E 93, 032118 – Published 11 March, 2016

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

Abstract

We study the probability p≡pη(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 all moments pm¯≃γm/t. From this, we extract the exact tail ∼ρ(p)/t of the probability distribution of the noncrossing probability at large time. The exact formula is compared to numerical simulations, with excellent agreement.

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Crossing probability for directed polymers in random media

Andrea De Luca and Pierre Le Doussal
Phys. Rev. E 92, 040102(R) (2015)

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Phys. Rev. E 93, 032118– Published 11 March, 2016
  • Received 25 November 2015

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