Harmonic and analytic functions on graphs (Q1402875)

Harmonic functions can be defined on every graph, while analytic functions can be defined on graphs embedded in orientable surfaces. Important properties of the ``true'' harmonic and analytic function can be carried over to the discrete setting. It is proved that a nonzero analytic function can vanish only on a very small connected piece. Consequently a simple local random process on embedded graphs is described.