Abstract: In 2017, Richard Stanley posed a fascinating question:
What is the asymptotically largest value of a principal specialization of a Schubert polynomial, and which permutation achieves it?
Although this problem remains out of reach in its full generality, we explore a related scenario by replacing Schubert polynomials with nonsymmetric Grothendieck polynomials. We define an ensemble of random permutations, where each permutation w\in S_n is assigned a probability proportional to the value of the corresponding Grothendieck polynomial. We then ask:
What do these "Grothendieck random permutations" typically look like for large n?
We show that these permutations have a limit shape (in the sense of permutons), and their fluctuations are governed by distributions from the Kardar-Parisi-Zhang universality class. Based on joint work with Alejandro Morales, Greta Panova, and Dmitriy Yeliussizov ([arXiv:2407.21653](https://arxiv.org/abs/2407.21653)).