Log-Gamma directed polymer with fixed endpoints via the replica Bethe Ansatz

Understand

We study the model of a discrete directed polymer (DP) on the square lattice with homogeneous inverse gamma distribution of site random Boltzmann weights, introduced by Seppalainen.

• The integer moments of the partition sum, $\overline{Z^n}$, are studied using a transfer matrix formulation, which appears as a generalization of the Lieb-Liniger quantum mechanics of bosons to discrete time and space.
• In the present case of the inverse gamma distribution the model is integrable in terms of a coordinate Bethe Ansatz, as discovered by Brunet.
• Using the Brunet-Bethe eigenstates we obtain an exact expression for the integer moments of $\overline{Z^n}$ for polymers of arbitrary lengths and fixed endpoint positions.