Séminaire Lotharingien de Combinatoire, 85B.34 (2021), 12 pp.

Pavel Galashin

Critical Varieties in the Grassmannian

Abstract. We introduce a family of spaces called \emph{critical varietieso}. The positive part of each critical variety is a subset of one of Postnikov's positroid cells inside the totally nonnegative Grassmannian. The combinatorics of positroid cells is captured by the dimer model on a planar bipartite graph G, and the critical variety is obtained by restricting to Kenyon's critical dimer model associated to a family of isoradial embeddings of G. This model is invariant under square/spider moves on G, and we give an explicit boundary measurement formula for critical varieties which does not depend on the choice of G. Special cases include critical electrical networks and Baxter's critical Z-invariant Ising model associated to rhombus tilings of polygons in the plane. In the case of regular polygons, our formula yields new simple expressions for response matrices of electrical networks and for correlation matrices of the Ising model.

Received: December 1, 2020. Accepted: March 1, 2021. Final version: April 29, 2021.

The following versions are available: