Fixed-Energy Harmonic Functions

Опубликовано 22 июня 2016, 0:22

We study the map from conductances to edge energies for harmonic functions on graphs with Dirichlet boundary conditions. We prove that for any compatible acyclic orientation and choice of energies there is a unique choice of conductances such that the associated harmonic function realizes those orientations and energies. For planar graphs there is a harmonic conjugate function; together these form the real and imaginary parts of a "fixed energy" analytic function. There is a geometric realization of the graph as a tiling of a plane region with fixed-area rectanges ("floorplans"). This is joint work with Aaron Abrams.