Dimers, webs, and local systems

Abstract

For a planar bipartite graph $\mathcal{G}$ equipped with a ${\mathrm{SL}}_n$-local system, we show that the determinant of the associated Kasteleyn matrix counts "$n$-multiwebs" (generalizations of $n$-webs) in $\mathcal{G}$, weighted by their web-traces. We use this fact to study random $n$-multiwebs in graphs on some simple surfaces.

Version 2: Same as published version