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We study the model of a discrete directed polymer (DP) on the square latticewith 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, whichappears as a generalization of the Lieb-Liniger quantum mechanics of bosons todiscrete time and space. In the present case of the inverse gamma distributionthe model is integrable in terms of a coordinate Bethe Ansatz, as discovered byBrunet. Using the Brunet-Bethe eigenstates we obtain an exact expression forthe integer moments of $\overline{Z^n}$ for polymers of arbitrary lengths andfixed endpoint positions. Although these moments do not exist for all integern, we are nevertheless able to construct a generating function which reproducesall existing integer moments, and which takes the form of a Fredholmdeterminant (FD). This suggests an analytic continuation via a Mellin-Barnestransform and we thereby propose a FD ansatz representation for the probabilitydistribution function (PDF) of $Z$ and its Laplace transform. In the limit ofvery long DP, this ansatz yields that the distribution of the free energyconverges to the GUE Tracy-Widom distribution up to a non-trivial average andvariance that we calculate. Our asymptotic predictions coincide with a resultby Borodin et al. based on a formula obtained by Seppalainen using the gRSKcorrespondence. In addition we obtain the dependence on the endpoint positionand the exact elastic coefficient at large time. We argue the equivalencebetween our formula and the one of Borodin et al. As we discuss, this open theway to explore the connections between quantum integrability and tropicalgeometry.

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