Discrete Riemann surfaces: linear discretization and its convergence

Abstract

We develop linear discretization of complex analysis, originally introduced by R. Isaacs, J. Ferrand, R. Duffin, and C. Mercat. We prove convergence of discrete period matrices and discrete Abelian integrals to their continuous counterparts. We also prove a discrete counterpart of the Riemann--Roch theorem. The proofs use energy estimates inspired by electrical networks.

27 pages, 4 figures. Major revision: Theorem 2.5, Lemma 2.7, their proofs, and the proof of Identity 3.1 corrected, new Sections 4.2, 4.3, 7.2, 7.3, and details to Section 5 added