This paper connects the theory of hyperplane arrangements to the theory oflinear resistor networks with fixed boundary voltages. Given a graph $G$, a set$partial Vsubsetneq V$, and a function $u:partial Vomathbb{R}$, our mainobject of study is the arrangement $mathcal{A}_{G,u}$ obtained from the realgraphic arrangement $mathcal{A}_G$ by fixing the coordinate $x_j$ to $u(j)$for all $jinpartial V$. First, fixed-energy harmonic functions in the senseof Abrams and Kenyon are shown to be critical points of master functions in thesense of Varchenko. Second, the basic graph-theoretic descriptions of$mathcal{A}_G$ are generalized to $mathcal{A}_{G,u}$. It is also proven thatthe arrangements $mathcal{A}_{G,u}$ are equivalent to the $psi$-graphicalarrangements introduced recently by Stanley.
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机译：本文将超平面布置理论连接到具有固定边界电压的线性电阻网络理论。给定图形$ g $，一个设置$ partial v subsetneq v $，以及函数$ u： partial v to mathbb {r} $，我们的mainObject是安排$ mathcal {a} _ {g，u} $从实图的安排$ mathcal {a} _g $获得，通过将协调$ x_j $固定为$ u（j）$以 partial v $。首先，Senseof Abrams和Kenyon中的固定能量谐波函数被证明是Varchenko的母函数的关键点。其次，$ mathcal {a} _g $的基本图形描述为$ mathcal {a} _ {g，u} $。还证明了安排$ mathcal {a} _ {g，u} $相当于斯坦利最近推出的$ psi $-traphicalArrangement。
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