Bethe anzats derivation of the Tracy-Widom distribution for one-dimensional directed polymers

The distribution function of the free energy fluctuations in one-dimensional directed polymers with $δ$-correlated random potential is studied by mapping the replicated problem to a many body quantum boson system with attractive interactions. Performing the summation over the entire spectrum of excited states the problem is reduced to the Fredholm determinant with the Airy kernel which is known to yield the Tracy-Widom distribution

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