The Fixed-$q$ Edge Limit of the $q$-Krawtchouk Ensemble

Abstract

We study the fixed-$q$ right-edge asymptotics of the ordinary $q$-Krawtchouk ensemble. For $0<q<1$, finite polynomial duality gives a direct proof of convergence of the correlation kernels to an explicit limiting kernel. The local Jacobi limit is a shifted continuous $q^{-1}$-Hermite expression with an indeterminate moment problem, so the limiting Jacobi expression does not determine a unique self-adjoint operator. A single spectral atom, asymptotically saturating the mass bound, selects the realization and yields strong-resolvent convergence. The limiting kernel is its positive spectral projection and admits spectral-series, Christoffel--Darboux, and partial-theta representations, and the associated dual operator agrees with a two-step spatial Markov operator of Borodin--Corwin. A companion $q>1$ limit is instead essentially self-adjoint, of Al-Salam--Carlitz~I type.