Random permutations from $q$-Demazure products

TL;DR

This paper investigates the $q$-deformation of the Demazure product model, demonstrating convergence to a deterministic permuton and resolving a prior conjecture about its limit.

Contribution

It proves the convergence of the $q$-deformed model to a specific permuton and explicitly identifies the limit, confirming a previous conjecture.

Findings

The permutation law converges to a deterministic permuton as $n$ approaches infinity.

The limiting permuton coincides with the $q=0$ case for an adjusted probability.

The conjecture from arXiv:2407.21653 about the limit was confirmed.

Abstract

We study the $q$-deformation of the Demazure product model from arXiv:2407.21653. Consider the longest element $w_0$ in $S_n$ written as a reduced word in simple transpositions. Independently delete each transposition with probability $1-p$ and apply the $q$-Demazure product to the remaining ones. We show that the law of the resulting permutation converges as $n \to \infty$ to a deterministic permuton, which coincides with the $q=0$ case studied in arXiv:2407.21653 for adjusted probability $p'=p(1-q)/(1-qp)$. This resolves Conjecture 1.13 from arXiv:2407.21653 and identifies the limiting permuton explicitly.