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

Spatial tightness at the edge of Gibbsian line ensembles