AI 中文总结
研究有限对角带状区域和半空间中对数伽马聚合物的平稳测度，通过独立\(\mathrm{Beta}_{II}\)随机变量建立其拉普拉斯变换表示，提出解析方法，证明带状区域平稳测度在宽度趋于无穷时收敛到半空间的，并推导半空间平稳测度拉普拉斯变换的轮廓积分公式。
AI 中文摘要
我们研究了有限对角带状区域和半空间中对数伽马聚合物的平稳测度。我们建立了带状区域平稳测度的相图，为此通过独立的\(\mathrm{Beta}_{II}\)随机变量开发了该平稳测度的拉普拉斯变换的一种表示。我们还提出了一种扩展巴拉昆德拉普拉斯变换轮廓积分表示的解析方法。我们证明，随着带状宽度趋于无穷，带状区域对数伽马聚合物的平稳测度收敛到半空间对数伽马聚合物的平稳测度。最后，我们推导了半空间对数伽马聚合物平稳测度拉普拉斯变换的轮廓积分公式。
英文摘要
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 $\mathrm{Beta}_{II}$ 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.
