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

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.

16 pages + 6 pages of appendices, 2 figures

References in corpus (11)

• Probability Distribution of the Free Energy of the Continuum Directed Random Polymer in 1+1 dimensions
• Burgers Turbulence
• An exact solution for the KPZ equation with flat initial conditions
• The KPZ equation with flat initial condition and the directed polymer with one free end
• Polynuclear growth on a flat substrate and edge scaling of GOE eigenvalues
• The One-dimensional KPZ Equation and the Airy Process
• Unbinding transition in semi-infinite two-dimensional localized systems
• Two-point generating function of the free energy for a directed polymer in a random medium
• From the sine-Gordon field theory to the Kardar-Parisi-Zhang growth equation
• Exact formulas for random growth with half-flat initial data
• The crossing probability for directed polymers in random media

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• Intermediate disorder directed polymers and the multi-layer extension of the stochastic heat equation
• Mutually avoiding paths in random media and largests eigenvalues of random matrices
• Distribution of Brownian coincidences
• The stochastic heat equation, 2D Toda equations and dynamics for the multilayer process
• A stationary model of non-intersecting directed polymers