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).

The geometric Burge correspondence and the partition function of polymer replicas / Bisi E; O'Connell N; Zygouras N. - In: SELECTA MATHEMATICA. - ISSN 1022-1824. - STAMPA. - 27:(2021), pp. 100.1-100.39. [10.1007/s00029-021-00712-8]

The geometric Burge correspondence and the partition function of polymer replicas

Bisi E;
2021

Abstract

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).
2021
27
1
39
Bisi E; O'Connell N; Zygouras N
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Utilizza questo identificatore per citare o creare un link a questa risorsa: https://hdl.handle.net/2158/1386112
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