Discrete Riemann Surfaces

TL;DR

This paper develops a comprehensive theory of Discrete Riemann Surfaces on cellular decompositions, establishing discrete analogs of classical concepts and exploring their properties and relationships.

Contribution

It introduces a rigorous framework for Discrete Riemann Surfaces, including discrete conformal structures and analogs of key theorems from the continuous theory.

Findings

Discrete period matrices are defined and analyzed.

Discrete Riemann's bilinear relations are established.

Connections between criticality and integrability are explored.

Abstract

We detail the theory of Discrete Riemann Surfaces. It takes place on a cellular decomposition of a surface, together with its Poincar\'e dual, equipped with a discrete conformal structure. A lot of theorems of the continuous theory follow through to the discrete case, we define the discrete analogs of period matrices, Riemann's bilinear relations, exponential of constant argument and series. We present the notion of criticality and its relationship with integrability.