Quicksilver Asymptotics and Pole Arrays for $q$-Painlevé II

We construct local families of solutions of the $q$-difference second Painlevé equation that vanish at infinity with a common algebraic expansion, parameterised by the amplitude of an exponentially small correction. Considering an appropriate boundary region, a formal expansion that retains two independently excited exponential perturbations is developed, and subsequent transseries summation predicts the location of simple poles. Adopting a $q$-theta function realisation of the exponential factors, we prove an asymptotic pole-location formula for a one-parameter family of true solutions. These poles approach the aforementioned boundaries where exponential behaviour fundamentally changes, providing an analogue of pole arrays near anti-Stokes directions in classical Painlevé asymptotics.