We derive exact formulas for the Laplace transforms of the partition functions. In the full-space this is expressed as a Fredholm determinant while in the half-space this is expressed as a Fredholm Pfaffian. Taking long-time asymptotics we show that the limiting free energy fluctuations scale with exponent 1/3 and are given by the GUE and GSE Tracy–Widom distributions. These formulas come from summing divergent moment generating functions, hence are not mathematically justified.
The primary purpose of this work is to present a mathematical perspective on the polymer replica method which is used to derive these results. In contrast to other replica method work, we do not appeal directly to the Bethe ansatz for the Lieb–Liniger model but rather utilize nested contour integral formulas for moments as well as their residue expansions.