固定$q$的$q$-Krawtchouk系综的边缘极限
The Fixed-$q$ Edge Limit of the $q$-Krawtchouk Ensemble
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中文总结 AI 辅助
本文研究$q$-Krawtchouk系综在固定$q$下的右边缘极限，通过有限多项式对偶性证明相关核收敛，并揭示极限算子的非唯一性及谱原子选择机制。
中文摘要 AI 辅助
我们研究普通$q$-Krawtchouk系综的固定$q$右边缘渐近行为。对于$0<q<1$，有限多项式对偶性给出了相关核收敛到显式极限核的直接证明。局部Jacobi极限是带有不定矩问题的平移连续$q^{-1}$-Hermite表达式，因此极限Jacobi表达式不能确定唯一的自伴算子。一个单独的谱原子，渐近地饱和质量界，选择实现并产生强预解收敛。极限核是其正谱投影，并具有谱级数、Christoffel--Darboux和部分theta表示，相关的对偶算子与Borodin--Corwin的两步空间马尔可夫算子一致。伴随的$q>1$极限本质上是自伴的，属于Al-Salam--Carlitz~I型。
英文摘要
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.
发表机构
- Beijing Institute of Mathematical Sciences and Applications (BIMSA)（北京数学与应用科学研究所）
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