  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 $\beta$.
