Toroidal Dimer Model and Temperley's Bijection

TL;DR

This paper explores the relationship between the toroidal dimer model and cycle rooted spanning forests via Temperley's bijection, analyzing their measure convergence and phase transition behavior as the torus size grows.

Contribution

It establishes the convergence of measures from the dimer model to spanning forests and characterizes the phase transition based on average height change.

Findings

Measure convergence to spanning forests or trees as size increases

Identification of a phase transition influenced by height change

Relation between height function and homology class in the model

Abstract

Temperley's bijection relates the toroidal dimer model to cycle rooted spanning forests ($CRSF$) on the torus. The height function of the dimer model and the homology class of $CRSF$ are naturally related. When the size of the torus tends to infinity, we show that the measure on $CRSF$ arising from the dimer model converges to a measure on (disconnected) spanning forests or spanning trees. There is a phase transition, which is determined by the average height change.