On the crystal limit of the q-difference sixth Painlev\'e equation

TL;DR

This paper investigates the behavior of the q-difference sixth Painlevé equation as q approaches zero, establishing the existence and explicit form of its crystal limit through the Riemann-Hilbert correspondence.

Contribution

It demonstrates the existence and explicit description of the crystal limit of the q-Painlevé VI equation's Riemann-Hilbert correspondence as q tends to zero.

Findings

Limit of the Riemann-Hilbert correspondence exists in the crystal limit.

The limiting map is bi-rational and explicitly described.

The study provides insights into the algebraic structure of the q-Painlevé VI equation at q=0.

Abstract

We consider the Riemann-Hilbert correspondence associated with the $q$-difference sixth Painlev\'e equation in the crystal limit, i.e. $q\rightarrow 0$, and show two main results. First, the limit of this generically highly transcendental mapping is shown to exist. Second, we show that the limiting map is bi-rational and describe it explicitly.