Log-gamma directed polymer with one free end via coordinate Bethe Ansatz

Pascal Grange

Published 2017-01-30Version 1

The discrete polymer model with random Boltzmann weights on sites with homogeneous inverse gamma distribution, introduced by Sepp\"al\"ainen, is studied in the case of a polymer with one fixed and one free end. The model with fixed end has been integrated by Thiery and Le Doussal, who used coordinate Bethe Ansatz techniques and proposed analytic-continuation prescriptions to represent the probability distribution of the free energy using the replica method, even though the moments of the partition sum do not exist at all orders due to the fat tail in the distribution of Boltzmann weights. To extend this approach to the polymer with one free end, we argue that the non-zero contribution to the partition sums in the thermodynamic limit is restricted to certain string states with a parity invariance in the space of rapidities. This situation is analogous to the case of the continuum polymer with one free end, related to the Kardar--Parisi--Zhang equation with flat boundary conditions and solved by Le Doussal and Calabrese. The expansion of the generating function in terms of numbers of strings can be transposed to the log-gamma polymer model, with the induced Fredholm determinant structure. We derive the large-time limit of the rescaled cumulative distribution function, and relate it to the Tracy--Widom distribution. The derivation is conjectural in the sense that it assumes completeness of a family of string states, and expressions of their norms, already useful in the fixed-end problem, and extends heuristically the order of moments of the partition sum to the complex plane.