The emergence of the electrostatic field as a Feynman sum in random tilings with holes

TL;DR

This paper demonstrates that the large-scale behavior of random lozenge tilings with holes converges to an electrostatic field, linking combinatorial tiling models to classical physics through a novel scaling limit analysis.

Contribution

It establishes a connection between the scaling limit of random tilings with holes and electrostatic fields, introducing a new interpretation of the emergent field as a Feynman sum.

Findings

The scaling limit of the tiling's average orientation field is an electrostatic field.

Holes act as charges with magnitudes determined by local tiling configurations.

The limit surface can be described as a sum of helicoids in the continuum.

Abstract

We consider random lozenge tilings on the triangular lattice with holes $Q_1,...,Q_n$ in some fixed position. For each unit triangle not in a hole, consider the average orientation of the lozenge covering it. We show that the scaling limit of this discrete field is the electrostatic field obtained when regarding each hole $Q_i$ as an electrical charge of magnitude equal to the difference between the number of unit triangles of the two different orientations inside $Q_i$. This is then restated in terms of random surfaces, yielding the result that the average over surfaces with prescribed height at the union of the boundaries of the holes is, in the scaling limit, a sum of helicoids.