Fixed-energy harmonic functions

TL;DR

This paper introduces the concept of enharmonic functions on graphs, exploring their properties, symmetries, and analogs to complex analysis, with implications for polygon tiling and potential theory.

Contribution

It establishes the existence and uniqueness of enharmonic functions for given energies and orientations, and develops a nonlinear analog of the Riemann mapping theorem.

Findings

Unique enharmonic functions exist for prescribed energies and orientations.

Galois group actions relate to symmetries of enharmonic functions.

Analogous nonlinear Cauchy-Riemann equations are derived for planar graphs.

Abstract

We study the map from conductances to edge energies for harmonic functions on finite 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. We call the associated function enharmonic. For rational energies and boundary data the Galois group of ${\mathbb Q}^{tr}$ (the totally real algebraic numbers) over ${\mathbb Q}$ permutes the enharmonic functions, acting on the set of compatible acyclic orientations. A consequence is the non-tileability of certain polygons by rational-area rectangles. For planar graphs there is an enharmonic conjugate function, together these form the real and imaginary parts of a "fixed energy" analytic function. In the planar scaling limit for ${\mathbb Z}^2$ (and the fixed south/west orientation), these functions satisfy a nonlinear analog of the Cauchy-Riemann equations, namely \begin{eqnarray*}u_xv_y &=& 1\\u_yv_x&=&-1.\end{eqnarray*} We give an analog of the Riemann mapping theorem for these functions, as well as a variational approach to finding solutions in both the discrete and continuous settings.