Grothendieck shenanigans: Permutons from pipe dreams via integrable probability

Morales A.H. Panova G. Petrov L. Yeliussizov D.
November 2025 Academic Press Inc.

Advances in Mathematics
2025 #480

We study random permutations corresponding to pipe dreams. Our main model is motivated by the Grothendieck polynomials with parameter β=1 arising in the 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 [Formula presented] and have the Tracy–Widom GUE distribution, which places this algebraic (K-theoretic) model into the Kardar–Parisi–Zhang universality class. As an application, we find the expected number of inversions in this random permutation, and contrast it with the case of non-reduced pipe dreams. Inspired by Stanleys question for the maximal value of principal specializations of Schubert polynomials, we resolve the analogous question for β=1 Grothendieck polynomials, and provide bounds for general β. This analysis uses a correspondence with the free fermion six-vertex model, and the frozen boundary of the Aztec diamond.

KPZ universality , Limit shape , Nonsymmetric Grothendieck polynomials , Permutons , Pipe dreams , Random permutations

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Université du Québec à Montréal, Montreal, QC, Canada
University of Massachusetts Amherst, Amherst, MA, United States
University of Southern California, Los Angeles, CA, United States
University of Virginia, Charlottesville, VA, United States
Kazakh–British Technical University, Almaty, Kazakhstan

Université du Québec à Montréal
University of Massachusetts Amherst
University of Southern California
University of Virginia
Kazakh–British Technical University

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