Localization in log-gamma polymers with boundaries

Consider the directed polymer in one space dimension in log-gamma environment with boundary conditions, introduced by Sepp{\"a}l{\"a}inen. In the equilibrium case, we prove that the end point of the polymer converges in law as the length increases, to a density proportional to the exponent of a zero-mean random walk. This holds without space normalization, and the mass concentrates in a neighborhood of the minimum of this random walk. We have analogous results out of equilibrium as well as for the middle point of the polymer with both ends fixed. The existence and the identification of the limit relies on the analysis of a random walk seen from its infimum.