Grothendieck Shenanigans: Permutons from pipe dreams via integrable probability

TL;DR

This paper explores the asymptotic behavior of permutations derived from reduced pipe dreams associated with Grothendieck polynomials, revealing Tracy-Widom fluctuations and connections to integrable probability and K-theory.

Contribution

It introduces a new probabilistic model based on Grothendieck polynomials, analyzes its limiting permuton, and links it to the KPZ universality class, also addressing open problems in permutation enumeration.

Findings

Limiting permuton described via TASEP mapping

Fluctuations follow Tracy-Widom GUE distribution

Provides bounds for principal specializations of Grothendieck polynomials

Abstract

We study random permutations arising from reduced pipe dreams. Our main model is motivated by Grothendieck polynomials with parameter $\beta=1$ arising in K-theory of the flag variety. The probability weight of a permutation is proportional to the principal specialization (setting all variables to 1) of the corresponding Grothendieck polynomial. By mapping this random permutation to a version of TASEP (Totally Asymmetric Simple Exclusion Process), we describe the limiting permuton and fluctuations around it as the order $n$ of the permutation grows to infinity. The fluctuations are of order $n^{\frac13}$ and have the Tracy-Widom GUE distribution, which places this algebraic (K-theoretic) model into the Kardar-Parisi-Zhang universality class. We also investigate non-reduced pipe dreams and make progress on a recent open problem on the asymptotic number of inversions of the resulting permutation. Inspired by Stanley's question for the maximal value of principal specializations of Schubert polynomials, we resolve the analogous question for $\beta=1$ Grothendieck polynomials, and provide bounds for general $\beta$.