Mathematics

Proof of a conjecture of Kenyon and Wilson on semicontiguous minors.

Tri Lai (University of Minnesota)

Abstract: The theory of "circular planar electrical networks" dates back to the 1990s with the work of Colin de Verdiere and Curtis, Ingerman, Moores, and Morrow. Associated with a circular planar electrical network is a response matrix, that measures the response of the network to potential applied at the nodes. Kenyon and Wilson showed how to test the well-connectivity of an electrical network with n nodes by checking the positivity of n(n-1)/2 central minors of the response matrix (arXiv:1411.7425). Their test was based on the fact that any contiguous minor of a matrix can be expressed as a Laurent polynomial in central minors. Moreover, the Laurent polynomial is the generating function of domino tilings of a weighted Aztec diamond. They conjectured that any semicontiguous minor can also be written in terms of domino tilings of a region on the square lattice. In this talk I will present a proof of the conjecture.