A note on consistency conditions on dimer models

TL;DR

This paper establishes the equivalence of different consistency conditions for non-degenerate dimer models, linking them through zigzag path properties and providing insights into their structural behavior.

Contribution

It demonstrates that the consistency conditions of Mozgovoy-Reineke and Gulotta are equivalent to a zigzag path condition, unifying different criteria in dimer model theory.

Findings

Equivalence of Mozgovoy-Reineke and Gulotta conditions for non-degenerate dimer models

Connection of these conditions to zigzag path properties

Application to the study of dimer model behavior under lattice polygon modifications

Abstract

We show that for a non-degenerate dimer model, both the first consistency condition of Mozgovoy and Reineke and the properly-orderedness condition of Gulotta are equivalent to a condition on zigzag paths, which goes back to Hanany and Vegh. The last condition is used in arXiv:0905.0059 to study the behavior of a dimer model under the operation of removing a vertex from the lattice polygon and taking the convex hull of the rest.