Conformal invariance of dimer heights on isoradial double graphs

TL;DR

This paper proves that the height distribution of perfect matchings on isoradial graphs converges to a conformally invariant Gaussian free field in the scaling limit, connecting discrete models to continuous conformal invariance.

Contribution

It establishes the conformal invariance of dimer height functions on isoradial graphs in the scaling limit, extending previous results to a broader class of graphs.

Findings

Height distribution converges to Gaussian free field

Conformal invariance holds in the scaling limit

Results apply to approximations of simply-connected domains

Abstract

An isoradial graph is a planar graph in which each face is inscribable into a circle of common radius. We study the 2-dimensional perfect matchings on a bipartite isoradial graph, obtained from the union of an isoradial graph and its interior dual graph. Using the isoradial graph to approximate a simply-connected domain bounded by a simple closed curve, by letting the mesh size go to zero, we prove that in the scaling limit, the distribution of height is conformally invariant and converges to a Gaussian free field.