几何Burge对应与聚合物复制的配分函数
The geometric Burge correspondence and the partition function of polymer replicas
AI总结：
本文提出了一种几何Burge对应的构造，展示了其与几何Robinson-Schensted-Knuth对应和Schützenberger反演的关系，并应用该模型证明了聚合物复制配分函数的分布为Whittaker测度。
摘要：
我们构建了一个几何提升的Burge对应，作为在通用Young图形数组上的局部birational映射的组合。我们建立了其与几何Robinson-Schensted-Knuth对应和几何Schützenberger反演的基本关系。我们还展示了几何Burge对应的若干性质，并将其专门化到对称输入数组的情况。特别是，我们的构造显示这种映射在log-log变量中是体积保持的。作为应用，我们考虑了两个给定长度的聚合物路径受限于相同终点的模型，即聚合物复制模型。我们证明了在log-gamma随机环境中，聚合物复制配分函数的分布是一个Whittaker测度，并推导出相应的Whittaker积分恒等式。对于特定的参数选择，我们注意到我们的模型与O'Connell、Seppäläinen和Zygouras（2014）研究的对称log-gamma聚合物之间存在分布恒等式。
英文摘要：
We construct a geometric lifting of the Burge correspondence as a composition of local birational maps on generic Young-diagram-shaped arrays. We establish its fundamental relation to the geometric Robinson-Schensted-Knuth correspondence and to the geometric Schützenberger involution. We also show a number of properties of the geometric Burge correspondence, specializing them to the case of symmetric input arrays. In particular, our construction shows that such a mapping is volume preserving in log-log variables. As an application, we consider a model of two polymer paths of given length constrained to have the same endpoint, known as polymer replica. We prove that the distribution of the polymer replica partition function in a log-gamma random environment is a Whittaker measure, and deduce the corresponding Whittaker integral identity. For a certain choice of the parameters, we notice a distributional identity between our model and the symmetric log-gamma polymer studied by O'Connell, Seppäläinen, and Zygouras (2014).
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