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Electrical response matrix of a regular \(2n\)-gon. (English) Zbl 1166.31001

Let \(P\) be a regular polygon with \(2n\) sides. The author considers a mixed Dirichlet–Neumann boundary value problem. Given constants \(v_1,v_2, \ldots, v_n\), there is a unique continuous function \(V\) on \(P\) which is harmonic on the interior, equals \(v_j\) on side \(2j\), and has zero normal derivative on the odd–numbered sides. Let \(E_k\) denote the normal derivative of \(V\) on side \(2k\). Since the outputs \(E_k\) depend linearly on the inputs \(v_j\), there is an \(n \times n\) matrix \(\Lambda\) mapping the vector \((v_1,v_2, \ldots, v_n)\) to the vector \((E_1,E_2, \ldots, E_n)\). The authors explicitly construct this matrix \(\Lambda\) and describe some connections with the limiting distributions of certain random spanning forests.

MSC:

31A25 Boundary value and inverse problems for harmonic functions in two dimensions
30C20 Conformal mappings of special domains
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
05C05 Trees
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References:

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