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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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This content will become publicly available on December 30, 2025
Random Fibonacci Words via Clone Schur Functions
We investigate positivity and probabilistic properties arising from the Young-Fibonacci lattice $$\mathbb{YF}$$, a 1-differential poset on binary words composed of 1's and 2's (known as Fibonacci words). Building on Okada's theory of clone Schur functions (Trans. Amer. Math. Soc. 346 (1994), 549-568), we introduce clone coherent measures on $$\mathbb{YF}$$ which give rise to random Fibonacci words of increasing length. Unlike coherent systems associated to classical Schur functions on the Young lattice of integer partitions, clone coherent measures are generally not extremal on $$\mathbb{YF}$$. Our first main result is a complete characterization of Fibonacci positive specializations - parameter sequences which yield positive clone Schur functions on $$\mathbb{YF}$$. We connect Fibonacci positivity with total positivity of tridiagonal matrices, Stieltjes moment sequences, and orthogonal polynomials in one variable from the ($$q$$-)Askey scheme. Our second family of results concerns the asymptotic behavior of random Fibonacci words derived from various Fibonacci positive specializations. We analyze several limiting regimes for specific examples, revealing stick-breaking-like processes (connected to GEM distributions), dependent stick-breaking processes of a new type, or discrete limits tied to the Martin boundary of the Young-Fibonacci lattice. Our stick-breaking-like scaling limits significantly extend the result of Gnedin-Kerov (Math. Proc. Camb. Philos. Soc. 129 (2000), no. 3, 433-446) on asymptotics of the Plancherel measure on $$\mathbb{YF}$$. We also establish Cauchy-like identities for clone Schur functions (with the right-hand side given by a quadridiagonal determinant), and construct and analyze models of random permutations and involutions based on Fibonacci positive specializations and a version of the Robinson-Schensted correspondence for $$\mathbb{YF}$$.
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- Award ID(s):
- 2153869
- PAR ID:
- 10611266
- Publisher / Repository:
- arXiv
- Date Published:
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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