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We study random permutations arising from reduced pipe dreams. Our main model is motivated by Grothendieck polynomials with parameter $$\beta=1$$ arising in K-theory of the flag variety. The probability weight of a permutation is proportional to the principal specialization (setting all variables to 1) of the corresponding Grothendieck polynomial. By mapping this random permutation to a version of TASEP (Totally Asymmetric Simple Exclusion Process), we describe the limiting permuton and fluctuations around it as the order $$n$$ of the permutation grows to infinity. The fluctuations are of order $$n^{\frac13}$$ and have the Tracy-Widom GUE distribution, which places this algebraic (K-theoretic) model into the Kardar-Parisi-Zhang universality class. We also investigate non-reduced pipe dreams and make progress on a recent open problem on the asymptotic number of inversions of the resulting permutation. Inspired by Stanley's question for the maximal value of principal specializations of Schubert polynomials, we resolve the analogous question for $$\beta=1$$ Grothendieck polynomials, and provide bounds for general $$\beta$$.
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Tilted biorthogonal ensembles, Grothendieck random partitions, and determinantal tests
We study probability measures on partitions based on symmetric Grothendieck polynomials. These deformations of Schur polynomials introduced in the K-theory of Grassmannians share many common properties. Our Grothendieck measures are analogs of the Schur measures on partitions introduced by Okounkov (Sel Math 7(1):57–81, 2001). Despite the similarity of determinantal formulas for the probability weights of Schur and Grothendieck measures, we demonstrate that Grothendieck measures are not determinantal point processes. This question is related to the principal minor assignment problem in algebraic geometry, and we employ a determinantal test first obtained by Nanson in 1897 for the 4 × 4 problem. We also propose a procedure for getting Nanson-like determinantal tests for matrices of any size n ≥ 4, which appear new for n ≥ 5. By placing the Grothendieck measures into a new framework of tilted biorthogonal ensembles generalizing a rich class of determinantal processes introduced by Borodin (Nucl Phys B 536:704–732, 1998), we identify Grothendieck random partitions as a cross-section of a Schur process, a determinantal process in two dimensions. This identification expresses the correlation functions of Grothendieck measures through sums of Fredholm determinants, which are not immediately suitable for asymptotic analysis. A more direct approach allows us to obtain a limit shape result for the Grothendieck random partitions. The limit shape curve is not particularly explicit as it arises as a cross-section of the limit shape surface for the Schur process. The gradient of this surface is expressed through the argument of a complex root of a cubic equation.
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- PAR ID:
- 10513377
- Publisher / Repository:
- Springer Nature
- Date Published:
- Journal Name:
- Selecta Mathematica
- Volume:
- 30
- Issue:
- 3
- ISSN:
- 1022-1824
- Page Range / eLocation ID:
- 56
- Subject(s) / Keyword(s):
- Grothendieck polynomials Determinantal processes Random partitions Limit shape Biorthogonal ensembles Principal minor assignment problem
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1007/s00029-024-00945-3
