Replica Bethe ansatz derivation of the Tracy-Widom distribution of the free energy fluctuations in one-dimensional directed polymers

Abstract

The distribution function of the free energy fluctuations in one-dimensional directed polymers with $\delta$-correlated random potential is studied by mapping the replicated problem to the $N$-particle quantum boson system with attractive interactions. We find the full set of eigenfunctions and eigenvalues of this many-body system and perform the summation over the entire spectrum of excited states. It is shown that in the thermodynamic limit the problem is reduced to the Fredholm determinant with the Airy kernel yielding the universal Tracy-Widom distribution, which is known to describe the statistical properties of the Gaussian unitary ensemble as well as many other statistical systems.

23 pages

References in corpus (7)

• Free-energy distribution of the directed polymer at high temperature
• Exact scaling functions for one-dimensional stationary KPZ growth
• Bethe anzats derivation of the Tracy-Widom distribution for one-dimensional directed polymers
• Correlation functions of the one-dimensional attractive Bose gas
• Optimal fluctuation approach to a directed polymer in a random medium
• Universal and non-universal tails of distribution functions in the directed polymer and KPZ problems
• Joint free energy distribution in the random directed polymer problem

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