  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.

Probability