The story started with a problem of Stanley from his 2017 paper “Some Schubert shenanigans”: what is the asymptotically maximal value of a principal specialization of a Schubert polynomial S_w, and what does the corresponding permutation w look like? While this question is still out of reach, we investigate its variant for nonsymmetric Grothendieck polynomials. We analyze the asymptotic behavior of random permutations whose probabilities are proportional to the Grothendieck polynomials using TASEP and tools from integrable probability. The typical behavior of these permutations is described by a specific permuton (a continuous analog of a permutation). We also examine a family of atypical layered permutations that achieve the same asymptotically maximal weight.

Based on the joint work with A.H. Morales, G. Panova, and D. Yeliussizov.

A random pipe dream and its coloring leading to the Grothendieck random permutation