Random Fibonacci Words - LaTeX Notes
Fibonacci words are words of 1’s and 2’s, graded by the total sum of the digits. They form a differential poset (YF) which is an estranged cousin of the Young lattice powering irreducible representations of the symmetric group. We introduce families of “coherent” measures on YF depending on many parameters, which come from the theory of clone Schur functions (Okada 1994). We characterize parameter sequences ensuring positivity of the measures, and we describe the large-scale behavior of some ensembles of random Fibonacci words. The subject has connections to total positivity of tridiagonal matrices, Stieltjes moment sequences, orthogonal polynomials from the (q-)Askey scheme, and residual allocation (stick-breaking) models.
Based on the joint work with J. Scott.
AIM workshop All roads to the KPZ universality class
Simulations of random permutations from the staircase reduced word
MATH 8380 • Random Matrices
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 $\mathbb{YF}$.
How we can use AI in Math work
These are the slides for the talk I gave at the University of Virginia on October 10, 2024 about how we can use AI in Math work.
How we can use AI in Math work - talk slides
Grothendieck Shenanigans
The story started with a problem of Stanley from his 2017 paper “Some Schubert shenanigans”: what is the asymptotically maximal value of a principal specialization of a Schubert polynomial S_w, and what does the corresponding permutation w look like? While this question is still out of reach, we investigate its variant for nonsymmetric Grothendieck polynomials. We analyze the asymptotic behavior of random permutations whose probabilities are proportional to the Grothendieck polynomials using TASEP and tools from integrable probability. The typical behavior of these permutations is described by a specific permuton (a continuous analog of a permutation). We also examine a family of atypical layered permutations that achieve the same asymptotically maximal weight.
Based on the joint work with A.H. Morales, G. Panova, and D. Yeliussizov.