Limit shape of perfect matchings on contracting bipartite graphs

TL;DR

This paper studies the asymptotic shape of perfect matchings on a class of contracting bipartite graphs, revealing a deterministic limit shape and multiple liquid regions, extending previous results to more general lattices.

Contribution

It introduces a method to determine the limit shape of perfect matchings on contracting bipartite graphs with periodic edge weights, generalizing prior work to broader lattice structures.

Findings

Established a deterministic limit shape in the scaling limit.

Proved the existence of multiple disconnected liquid regions.

Extended results to more general contracting square-hexagon lattices.

Abstract

We consider random perfect matchings on a general class of contracting bipartite graphs by letting certain edge weights be 0 on the contracting square-hexagon lattice in a periodic way. We obtain a deterministic limit shape in the scaling limit. The results can also be applied to prove the existence of multiple disconnected liquid regions for all the contracting square-hexagon lattices with certain edge weights, extending the results proved in [13] for contracting square-hexagon lattices where the number of square rows in each period is either 0 or 1.