Attempted Bethe ansatz solution for one-dimensional directed polymers in random media

Abstract

We study the statistical properties of one-dimensional directed polymers in a short-range random potential by mapping the replicated problem to a many body quantum boson system with attractive interactions. We find the full set of eigenvalues and eigenfunctions of the many-body system and perform the summation over the entire spectrum of excited states. The analytic continuation of the obtained exact expression for the replica partition function from integer to non-integer replica parameter N turns out to be ambiguous. Performing the analytic continuation simply by assuming that the parameter N can take arbitrary complex values, and going to the thermodynamic limit of the original directed polymer problem, we obtain the explicit universal expression for the probability distribution function of free energy fluctuations.

32 pages, 1 figure

References in corpus (4)

• Exact scaling functions for one-dimensional stationary KPZ growth
• Correlation functions of the one-dimensional attractive Bose gas
• Optimal fluctuation approach to a directed polymer in a random medium
• Joint free energy distribution in the random directed polymer problem

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