Limit shape and arctic curve for uniformly random RBPDs

Establish a deterministic limit shape for the height function of uniformly random reduced bumpless pipe dreams of size n, with four frozen regions and a liquid region separated by an arctic curve, and prove that the southeast arc of this arctic curve coincides with the support of the singular component of the Schubert permuton.

Background

The discrete mixed derivative of the averaged height function highlights a liquid region where all six tile types coexist, surrounded by frozen regions. Numerical experiments suggest a non–quarter-turn-symmetric arctic curve, unlike certain integrable cases (e.g., Grothendieck/β=1 or uniform six-vertex).

The conjecture links the arctic curve’s southeast boundary to the singular support of the Schubert permuton, and a proposition in the paper provides a partial implication under a strong frozen-region assumption.