Discrete period matrices and related topics

TL;DR

This paper explores discrete analogs of classical Riemann surface concepts, demonstrating convergence of discrete period matrices to their continuous counterparts through critical map sequences.

Contribution

It introduces discrete period matrices and proves their convergence to continuous matrices using critical map sequences.

Findings

Discrete period matrices are well-defined and relate to classical matrices.

Convergence of discrete to continuous period matrices is established.

New discrete analogs of Riemann's bilinear relations are developed.

Abstract

We continue our investigation of Discrete Riemann Surfaces with the discussion of the discrete analogs of period matrices, Riemann's bilinear relations, exponential of constant argument, series and electrical moves. We show that given a refining sequence of critical maps, the discrete period matrix converges to the continuous one.