Scaling limits of discrete holomorphic functions

TL;DR

This paper proves the convergence of discrete holomorphic functions to continuous ones on square lattices by introducing new concepts and establishing a discrete integral formula, solving a longstanding open problem in discrete complex analysis.

Contribution

It introduces new concepts of discrete surface measure and outer normal vector, and establishes a discrete Cauchy-Pompeiu integral formula for the first time.

Findings

Uniform convergence of discrete holomorphic functions to continuous functions.

Convergence holds up to second order derivatives.

Results apply to standard square lattices.

Abstract

One of the most natural and challenging issues in discrete complex analysis is to prove the convergence of discrete holomorphic functions to their continuous counterparts. This article is to solve the open problem in the general setting. To this end we introduce new concepts of discrete surface measure and discrete outer normal vector and establish the discrete Cauchy-Pompeiu integral formula, \begin{eqnarray*} f(\zeta)=\displaystyle{\int_{\partial B^h}} \mathcal{K}^h(z,\zeta) f(z)dS^h(z)+\displaystyle{\int_{B^h}} E^h(\zeta-z) \partial_{\bar z}^h f (z)dV^h(z),\end{eqnarray*} which results in the uniform convergence of the scaling limits of discrete holomorphic functions up to second order derivatives in the standard square lattices.