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 represents its positive spectral projection and admits spectral-series, Christoffel--Darboux, and partial-theta representations. After a ground-state transformation, the associated dual operator agrees with a two-step spatial Markov operator of Borodin--Corwin. For q>1, the limiting Jacobi operator is of Al-Salam--Carlitz~I type and is essentially self-adjoint.

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BibTeX

@misc{nikitin2026fixedq,
  title = {{The Fixed-$q$ Edge Limit of the $q$-Krawtchouk Ensemble}},
  author = {Pavel Nikitin},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.24149},
  eprint = {2609.24149},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.24149},
}

RIS

TY  - UNPB
TI  - The Fixed-q Edge Limit of the q-Krawtchouk Ensemble
AU  - Pavel Nikitin
PY  - 2026
DA  - 2026/10/23
PB  - arXiv
DO  - 10.48550/arxiv.2609.24149
UR  - https://arxiv.org/abs/2609.24149
AB  - 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 represents its positive spectral projection and admits spectral-series, Christoffel--Darboux, and partial-theta representations. After a ground-state transformation, the associated dual operator agrees with a two-step spatial Markov operator of Borodin--Corwin. For q>1, the limiting Jacobi operator is of Al-Salam--Carlitz~I type and is essentially self-adjoint.
ER  -

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