Fixed-Energy Harmonic Functions


July 22, 2015


Rick Kenyon


Brown University


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.


Rick Kenyon

Richard Kenyon is the William R. Kenan Jr. University Professor of Mathematics at Brown University. He received his PhD in 1990, advised by Bill Thurston at Princeton. His research has dealt with the interface between statistical mechanics, probability and discrete conformal geometry. In particular, his works on Local statistics of lattice dimers and loop erased random walks have had a major on the large field studying Gibbs distributions of combinatorial configurations. Kenyon is a recipient of the Rollo Davidson prize (2001), the Loeve prize (2007) and was a Clay Senior Fellow in 2012. His publications are available at