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We study random permutations corresponding to pipe dreams. Our main model is motivated by the Grothendieck polynomials with parameter ß = 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 polyno- mial. 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$ 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 B = 1 Grothendieck polynomials, and provide bounds for general B. This analysis uses a correspondence with the free fermion six-vertex model, and the frozen boundary of the Aztec diamond. 

The number of standard Young tableaux of a skew shape $$\lambda/\mu$$ can be computed as a sum over excited diagrams inside $$\lambda$$. Excited diagrams are in bijection with certain lozenge tilings, with flagged semistandard tableaux and also nonintersecting lattice paths inside $$\lambda$$. We give two new proofs of a multivariate generalization of this formula, which allow us to extend the setup beyond standard Young tableaux and the underlying Schur symmetric polynomials. The first proof uses multiple contour integrals. The second one interprets excited diagrams as configurations of a six-vertex model at a free fermion point, and derives the formula for the number of standard Young tableaux of a skew shape from the Yang-Baxter equation. 

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 $$\beta$$.