Elliptic asymptotic behaviour of $q$-Painlevé transcendents

The discrete Painlevé equations have mathematical properties closely related to those of the differential Painlevé equations. We investigate the appearance of elliptic functions as limiting behaviours of $q$-Painlevé transcendents, analogous to the asymptotic theory of classical Painlevé transcendents. We focus on the $q$-difference second Painlevé equation in the asymptotic regime $\mathrm{\mid }q-1\mathrm{\mid }\ll 1$, showing that generic leading-order behaviour is given in terms of elliptic functions and that the slow modulation in this behaviour is approximated in terms of complete elliptic integrals.