Stationary measures for log-gamma polymer on a strip and in half-space

We study stationary measures of the log-gamma polymer on a finite diagonal strip and in a half-space. We establish the phase diagram for the stationary measure on the strip. To this end we develop a representation of {the Laplace transform of} this stationary measure by independent ${\mathrm{B}\mathrm{e}\mathrm{t}\mathrm{a}}_{II}$ random variables. We also present an analytic approach that extends Barraquand’s contour integral representation of the Laplace transform. We prove that, as the strip width tends to infinity, the stationary measure of the log-gamma polymer on the strip converges to the stationary measure of the half-space log-gamma polymer. Finally, we derive a contour integral formula for the Laplace transform of the stationary measure of the half-space log-gamma polymer.