Scaling limit of domino tilings on a pentagonal domain

TL;DR

This paper analyzes the scaling limit of domino tilings on a pentagonal domain derived from the Aztec diamond, revealing a third-order phase transition related to geometric tuning of the domain.

Contribution

It computes the scaling limit of a non-local correlation function for domino tilings on a pentagon-shaped domain, connecting geometric parameters to phase transitions.

Findings

Identifies the scaling limit of a correlation function for domino tilings on a pentagon.

Discovers a third-order phase transition when the pentagon's side is tangent to the arctic ellipse.

Links geometric tuning to phase transition phenomena in domino tilings.

Abstract

We consider the six-vertex model at its free-fermion point with domain wall boundary conditions, which is equivalent to random domino tilings of the Aztec diamond. We compute the scaling limit of a particular non-local correlation function, essentially equivalent to the partition function for the domino tilings of a pentagon-shaped domain, obtained by cutting away a triangular region from a corner of the initial Aztec diamond. We observe a third-order phase transition when the geometric parameters of the obtained pentagonal domain are tuned to have the fifth side exactly tangent to the arctic ellipse of the corresponding initial model.