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 $\beta =1$ 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 $n^{\frac{1}{3}}$ and have the Tracy–Widom GUE distribution, which places this algebraic (K-theoretic) model into the Kardar–Parisi–Zhang universality class. 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 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 β. This analysis uses a correspondence with the free fermion six-vertex model, and the frozen boundary of the Aztec diamond.