Planar $3$-webs and the boundary measurement matrix

We compute connection probabilities for reduced $3$-webs in the triple-dimer model on circular planar graphs using the boundary measurement matrix (reduced Kasteleyn matrix). As one application we compute several "${\text{SL}}_3$ generalizations’’ of the Lindstrøm-Gessel-Viennot theorem, for “parallel” webs and for honeycomb webs. We also apply our results to the scaling limit of the dimer model in a planar domain, giving conformally invariant expressions for reduced web probabilities.