The asymptotic behaviour of the discrete holomorphic map $Z^a$ via the Riemann-Hilbert method

TL;DR

This paper analyzes the long-term behavior of a discrete version of the holomorphic map z^a using the Riemann-Hilbert method, confirming a conjecture from 2000 about its asymptotics.

Contribution

It applies the Riemann-Hilbert approach and Deift-Zhou steepest descent method to rigorously establish asymptotic formulas for the discrete holomorphic map z^a, validating a longstanding conjecture.

Findings

Proved the asymptotic formula for the discrete holomorphic map z^a

Validated the conjecture from 2000 about the map's asymptotics

Demonstrated the effectiveness of the Riemann-Hilbert method in discrete complex analysis

Abstract

We study the asymptotic behavior of the discrete analogue of the holomorphic map $z^a$. The analysis is based on the use of the Riemann-Hilbert approach. Specifically, using the Deift-Zhou nonlinear steepest descent method we prove the asymptotic formulae which was conjectured in 2000 by the first co-author and S.I.~Agafonov.