Crossover in the log-gamma polymer from the replica coordinate Bethe Ansatz

The coordinate Bethe Ansatz solution of the log-gamma polymer is extended to boundary conditions with one fixed end and the other attached to one half of a one-dimensional lattice. The large-time limit is studied using a saddle-point approximation,and the cumulative distribution function of the rescaled free energy of a long polymer is expressed as a Fredholm determinant. Scaling limits of the kernel are identified, leading to a crossover from the GUE to the GOE Tracy--Widom distributions. The continuum limit reproduces the crossover from droplet to flat initial conditions of the Kardar--Parisi--Zhang equation.