Proof of a conjecture of Kenyon and Wilson on semicontiguous minors

JOURNAL OF COMBINATORIAL THEORY SERIES A

卷:161

Article

Lai, Tri¹ 

[1] Univ Nebraska, Dept Math, Lincoln, NE 68588 USA

关键词: Perfect matchings;    Domino tilings;    Dual graph;    Graphical condensation;    Electrical networks;    Response matrix;    Aztec diamonds;   

DOI  :  10.1016/j.jcta.2018.07.008

来源: Elsevier

【 摘 要 】

Kenyon and Wilson showed how to test if a circular planar electrical network with n nodes is well-connected by checking the positivity of (n 2) 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 things 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. (C) 2018 Elsevier Inc. All rights reserved.

【 授权许可】

Free