Tilted biorthogonal ensembles, Grothendieck random partitions, and determinantal tests Article Swipe
Svetlana Gavrilova
,
Leonid Petrov
·
YOU?
·
· 2024
· Open Access
·
· DOI: https://doi.org/10.1007/s00029-024-00945-3
YOU?
·
· 2024
· Open Access
·
· DOI: https://doi.org/10.1007/s00029-024-00945-3
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 $$4\times 4$$ problem. We also propose a procedure for getting Nanson-like determinantal tests for matrices of any size $$n\ge 4$$ , which appear new for $$n\ge 5$$ . 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.
Related Topics
Concepts
Biorthogonal system
Mathematics
Combinatorics
Determinantal point process
Statistical physics
Pure mathematics
Random matrix
Computer science
Physics
Artificial intelligence
Quantum mechanics
Eigenvalues and eigenvectors
Wavelet
Wavelet transform
Metadata
- Type
- article
- Language
- en
- Landing Page
- https://doi.org/10.1007/s00029-024-00945-3
- https://link.springer.com/content/pdf/10.1007/s00029-024-00945-3.pdf
- OA Status
- hybrid
- References
- 70
- Related Works
- 10
- OpenAlex ID
- https://openalex.org/W4399495873
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https://openalex.org/W4399495873Canonical identifier for this work in OpenAlex
- DOI
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https://doi.org/10.1007/s00029-024-00945-3Digital Object Identifier
- Title
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Tilted biorthogonal ensembles, Grothendieck random partitions, and determinantal testsWork title
- Type
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articleOpenAlex work type
- Language
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enPrimary language
- Publication year
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2024Year of publication
- Publication date
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2024-06-10Full publication date if available
- Authors
-
Svetlana Gavrilova, Leonid PetrovList of authors in order
- Landing page
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https://doi.org/10.1007/s00029-024-00945-3Publisher landing page
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https://link.springer.com/content/pdf/10.1007/s00029-024-00945-3.pdfDirect link to full text PDF
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hybridOpen access status per OpenAlex
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https://link.springer.com/content/pdf/10.1007/s00029-024-00945-3.pdfDirect OA link when available
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| corresponding_institution_ids | https://openalex.org/I51556381 |
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| citation_normalized_percentile.is_in_top_10_percent | False |