Electrical networks and hyperplane arrangements

Bob Lutz

Published 2017-09-05Version 1

This paper connects the theory of hyperplane arrangements to the theory of linear resistor networks with fixed boundary voltages. Given a graph $G$, a set $\partial V\subsetneq V$, and a function $u:\partial V\to\mathbb{R}$, our main object of study is the arrangement $\mathcal{A}_{G,u}$ obtained from the real graphic arrangement $\mathcal{A}_G$ by fixing the coordinate $x_j$ to $u(j)$ for all $j\in\partial V$. First, fixed-energy harmonic functions in the sense of Abrams and Kenyon are shown to be critical points of master functions in the sense of Varchenko. Second, the basic graph-theoretic descriptions of $\mathcal{A}_G$ are generalized to $\mathcal{A}_{G,u}$. It is also proven that the arrangements $\mathcal{A}_{G,u}$ are equivalent to the $\psi$-graphical arrangements introduced recently by Stanley.