  We introduce a random walk in random environment associated to an underlying
directed polymer model in $1+1$ dimensions. This walk is the positive
temperature counterpart of the competition interface of percolation and arises
as the limit of quenched polymer measures. We prove this limit for the exactly
solvable log-gamma polymer, as a consequence of almost sure limits of ratios of
partition functions. These limits of ratios give the Busemann functions of the
log-gamma polymer, and furnish centered cocycles that solve a variational
formula for the limiting free energy. Limits of ratios of point-to-point and
point-to-line partition functions manifest a duality between tilt and velocity
that comes from quenched large deviations under polymer measures. In the
log-gamma case, we identify a family of ergodic invariant distributions for the
random walk in random environment.
