Random Fibonacci Words via Clone Schur Functions

TL;DR

This paper explores the probabilistic and positivity properties of Fibonacci words within a Young--Fibonacci lattice, connecting clone Schur functions to various mathematical structures and analyzing asymptotic behaviors of associated random words.

Contribution

It characterizes Fibonacci positive specializations of clone Schur functions, links positivity to multiple mathematical areas, and extends asymptotic analysis of random Fibonacci words.

Findings

Complete characterization of Fibonacci positive specializations

Connections between positivity, total positivity, and orthogonal polynomials

New limit laws for random Fibonacci words and asymptotics of the Young--Fibonacci lattice

Abstract

We study positivity and probabilistic properties arising from the Young--Fibonacci lattice $\mathbb{YF}$, a 1-differential poset on binary (Fibonacci) words of 1's and 2's, graded by digit sum. Building on Okada's theory of clone Schur functions (Trans. Amer. Math. Soc. 346 (1994), 549--568), we define clone coherent measures on $\mathbb{YF}$ that generate random Fibonacci words of increasing length; unlike for the Young lattice (powered by the classical Schur functions), 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 $\mathbb{YF}$. Second, we connect Fibonacci positivity with: (i) total positivity of tridiagonal matrices; (ii) Stieltjes moment sequences; (iii) the combinatorics of set partitions; and (iv) families of univariate orthogonal polynomials from the (q-)Askey scheme. We further link moment sequences of orthogonal polynomials to combinatorial structures on Fibonacci words, a connection that may be of independent interest. Third, we analyze scaling limits of the induced random words, obtaining stick-breaking-type limits (linked to GEM laws), new dependent stick-breaking limits, and limits supported on the discrete part of the Martin boundary of $\mathbb{YF}$. These results significantly extend the asymptotics of the Plancherel measure on $\mathbb{YF}$ proved by Gnedin--Kerov (Math. Proc. Camb. Philos. Soc. 129 (2000), 433--446). Finally, we prove Cauchy-type identities for clone Schur functions with quadridiagonal-determinant right-hand side (in contrast to the product form for classical Schur functions), and construct models of random permutations and involutions from Fibonacci-positive specializations together with a Robinson--Schensted correspondence adapted to $\mathbb{YF}$.