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The geometric Burge correspondence and the partition function of polymer replicas

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Bisi, Elia, O’Connell, Neil and Zygouras, Nikos (2021) The geometric Burge correspondence and the partition function of polymer replicas. Selecta Mathematica, New Series, 27 . 100. doi:10.1007/s00029-021-00712-8

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Official URL: https://doi.org/10.1007/s00029-021-00712-8

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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 fun- damental 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 iden- tity between our model and the symmetric log-gamma polymer studied by O’Connell, Seppäläinen, and Zygouras (2014).

Item Type: Journal Article
Subjects: Q Science > QA Mathematics
Divisions: Faculty of Science, Engineering and Medicine > Science > Mathematics
Library of Congress Subject Headings (LCSH): Combinatorial analysis, Symmetric functions
Journal or Publication Title: Selecta Mathematica, New Series
Publisher: Birkhauser Verlag AG
ISSN: 1022-1824
Official Date: 13 October 2021
Dates:
DateEvent
13 October 2021Published
16 September 2021Accepted
Volume: 27
Article Number: 100
DOI: 10.1007/s00029-021-00712-8
Status: Peer Reviewed
Publication Status: Published
Access rights to Published version: Open Access
RIOXX Funder/Project Grant:
Project/Grant IDRIOXX Funder NameFunder ID
EP/R024456/1[EPSRC] Engineering and Physical Sciences Research Councilhttp://dx.doi.org/10.13039/501100000266
669306European Research Councilhttp://dx.doi.org/10.13039/501100000781
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