The geometric Burge correspondence and the partition function of polymer replicas

We construct a geometric lifting of the Burge correspondence as composition local birational maps on generic Young-diagram-shaped arrays. establish its fundamental relation to Robinson-Schensted-Knuth and Sch\"utzenberger involution. also show number properties correspondence, specializing them case symmetric input In particular, our construction shows that such mapping is volume preserving in log-log variables. As an application, we consider model two polymer paths given length constrained have same endpoint, known replica. prove distribution replica partition function log-gamma random environment Whittaker measure, deduce corresponding integral identity. For certain choice parameters, notice distributional identity between studied by O'Connell, Sepp\"al\"ainen, Zygouras (2014).