Proof of a conjecture of Kenyon and Wilson on semicontiguous minors

Abstract

Kenyon and Wilson showed how to test if a circular planar electrical network with $n$ nodes is well-connected by checking the positivity of $\left(\genfrac{}{}{0ex}{}{n}{2}\right)$ central minors of the response matrix. Their test is based on the fact that any contiguous minor of a matrix can be expressed as a Laurent polynomial in the central minors. Moreover, the Laurent polynomial is the generating function of domino tilings of a weighted Aztec diamond. They conjectured that a larger family of minors, semicontiguous minors, can also be written in terms of domino tilings of a region on the square lattice. In this paper, we present a proof of the conjecture.

Final version: 33 pages. To appear in Journal of Combinatorial Theory, Series A

References in corpus (4)

• Boundary Partitions in Trees and Dimers
• Condensation of Determinants
• Graphical Condensation Generalizations Involving Pfaffians and Determinants
• Electrical Lie Algebra of Classical Types