2024-07-31-Grothendieck-shenanigans.md

layout	post
title	Grothendieck Shenanigans: Permutons from Pipe Dreams via Integrable Probability
arXiv	2407.21653 [math.PR]
coauthors	name web Alejandro H. Morales https://professeurs.uqam.ca/professeur/morales_borrero.alejandro/ name web Greta Panova https://sites.google.com/usc.edu/gpanova/home?authuser=0 name Leonid Petrov name web Damir Yeliussizov http://www.damir.xyz/
comments	false
categories	blog math paper
selected	true
journal-ref	Advances in Mathematics, 480 (2025), Part C, 110510.
journal-web	https://www.sciencedirect.com/science/article/pii/S0001870825004086
journal-ref-extra	FPSAC 2025 poster
journal-year	2025
published	true
image	__STORAGE_URL__/img/papers/grothendieck-shenanigans.png
image-alt	Reduction of a pipe dream leading to the permutation w(D) = 241653.
show-date	true
pdf	45-grothendieck-shenanigans-permutons.pdf
post-pdf	true
pages	50
cv-number	45
simulations	simulations/2025-01-26-grothendieck-shenanigans/

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 $n^{\frac{1}{3}}$ 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 $\beta$.

FPSAC poster version