Harmonic and analytic functions on graphs

Harmonic and analytic functions have natural discrete analogues. Harmonic functions can be defined on every graph, while analytic functions (or, more precisely, holomorphic forms) can be defined on graphs embedded in orientable surfaces. Many important properties of the “true” harmonic and analytic functions can be carried over to the discrete setting. We prove that a nonzero analytic function can vanish only on a very small connected piece. As an application, we describe a simple local random process on embedded graphs, which have the property that observing them in a small neighborhood of a node through a polynomial time, we can infer the genus of the surface.