Convergence to a Schubert permuton

Establish that the Schubert measure on permutations, defined by P(w) ∝ 𝛶_w with 𝛶_w = 𝔖_w(1^n), converges as n→∞ to a deterministic permuton supported inside a cone and possessing a singular component along its southeast boundary curve.

Background

The Schubert measure arises by weighting permutations w by the number of reduced bumpless pipe dreams with boundary permutation w. For Grothendieck polynomials at β=1, the analogous measure is known to converge to a permuton with a singular boundary component.

Simulations here suggest a similar phenomenon for Schubert polynomials, motivating a conjectural permuton limit with characteristic geometric features.

References

The average permutation matrix we observe for the Schubert measure is visually similar (\Cref{fig:permuton_n100}), motivating us to make the following conjecture: Conjecture [Convergence to the Schubert permuton] The random permutations eq:schubert_measure converge, as $n\to\infty$, to a deterministic permuton $\mu$ on $[0,1]2$ (which we call the Schubert permuton) supported inside a cone and having a singular component along the southeast boundary curve.

— Computation and sampling for Schubert specializations  (2603.20104 - Anderson et al., 20 Mar 2026) in Section 6.1