  Consider a sequence of Gibbsian line ensembles, whose lowest labeled curves
(i.e., the edge) have tight one-point marginals. Then, given certain technical
assumptions on the nature of the Gibbs property and underlying random walk
measure, we prove that the entire spatial process of the edge is tight. We then
apply this black-box theory to the log-gamma polymer Gibbsian line ensemble
which we construct. The edge of this line ensemble is the transversal free
energy process for the polymer, and our theorem implies tightness with the
ubiquitous KPZ class $2/3$ exponent, as well as Brownian absolute continuity of
all the subsequential limits.
  A key technical innovation which fuels our general result is the construction
of a continuous grand monotone coupling of Gibbsian line ensembles with respect
to their boundary data (entrance and exit values, and bounding curves). {\em
Continuous} means that the Gibbs measure varies continuously with respect to
varying the boundary data, {\em grand} means that all uncountably many boundary
data measures are coupled to the same probability space, and {\em monotone}
means that raising the values of the boundary data likewise raises the
associated measure. This result applies to a general class of Gibbsian line
ensembles where the underlying random walk measure is discrete time, continuous
valued and log-convex, and the interaction Hamiltonian is nearest neighbor and
convex.
