Alternative scalings yielding distinct q→0 Riemann–Hilbert limits for q-Painlevé VI

Determine whether there exist alternative scalings of the independent variable t and the parameter q that, as q → 0, produce different limiting Riemann–Hilbert problems for the q-difference sixth Painlevé equation, and characterize the resulting limiting correspondences; in particular, analyze ultra-discrete scalings such as t = e^{-T/epsilon} and q = e^{-Q/epsilon} with epsilon → 1 and fixed T and Q.

Background

The main results are obtained under the straightforward crystal limit q → 0 with fixed t, leading to an explicit isomorphism between the initial value surface and a Segre surface. The paper suggests that other q → 0 limits may be meaningful when t is scaled simultaneously with q.

The authors specifically point to ultra-discrete scalings of t and q, which are known to yield ultra-discrete Painlevé equations, and propose investigating whether such scalings induce distinct and well-posed Riemann–Hilbert problems and monodromy manifolds in the limit. Establishing these would extend the crystal-limit framework to broader asymptotic regimes.