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Weak noise theory of the O'Connell-Yor polymer as an integrable discretization of the nonlinear Schrödinger equation
Alexandre Krajenbrink1, Pierre Le Doussal2
1Quantinuum, Terrington House, 13-15 Hills Road, Cambridge CB2 1NL, United Kingdom and Le Lab Quantique, 58 rue d'Hauteville, 75010 Paris, France.
Abstract:
We investigate and solve the weak noise theory for the semidiscrete O'Connell-Yor directed polymer. In the large deviation regime, the most probable evolution of the partition function obeys a classical nonlinear system which is a nonstandard discretization of the nonlinear Schrödinger equation with mixed initial-final conditions. We show that this system is integrable and find its general solution through an inverse scattering method and a non-standard Fredholm determinant framework that we develop. This allows us to obtain the large deviation rate function of the free energy of the polymer model from its conserved quantities and to study its convergence to the large deviations of the Kardar-Parisi-Zhang equation. Our model also degenerates to the classical Toda chain, which further substantiates the applicability of our Fredholm framework.
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