Séminaire Lotharingien de Combinatoire, 93B.28 (2025), 12 pp.
Alejandro H. Morales, Greta Panova, Leonid Petrov and Damir Yeliussizov
Grothendieck Shenanigans:
Permutons from Pipe Dreams via Integrable Probability
Abstract.
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 its
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 n1/3 and have the Tracy-Widom GUE distribution,
which places this algebraic (K-theoretic) model
into the Kardar-Parisi-Zhang universality class.
Inspired by
Stanley's 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.
Received: November 15, 2024.
Accepted: February 15, 2025.
Final version: April 1, 2025.
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