Integrable Contour Kernels in Discrete $β=1,4$ Ensembles, Universality and Kuznetsov Multipliers
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915612058451968 |
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| author | Tierz, Miguel |
| author_facet | Tierz, Miguel |
| contents | We obtain explicit double-contour representations for the correlation kernels of the discrete orthogonal ($β=1$) and symplectic ($β=4$) random matrix ensembles with Meixner, Charlier, and Krawtchouk weights. A single Cauchy--difference--quotient composition identity expresses all $β=1,4$ blocks in terms of the projection kernel and bounded rational multipliers. From these formulas we give short steepest-descent proofs of bulk and edge universality (sine/Airy/Bessel) with uniform error control, an explicit Meixner$\to$Laguerre hard-edge crossover, and a first $A^{-1}$ correction that follows directly from the integrable structure. Finally, we show that Archimedean Kuznetsov tests splice into the Pfaffian kernels by a bounded holomorphic symbol acting in the contour variable; the symbol enters only through the same Cauchy difference--quotient, so the leading sine/Airy/Bessel limits persist and the $A^{-1}$ term again comes from linearizing at the saddle(s). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_08356 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Integrable Contour Kernels in Discrete $β=1,4$ Ensembles, Universality and Kuznetsov Multipliers Tierz, Miguel Mathematical Physics Number Theory Probability We obtain explicit double-contour representations for the correlation kernels of the discrete orthogonal ($β=1$) and symplectic ($β=4$) random matrix ensembles with Meixner, Charlier, and Krawtchouk weights. A single Cauchy--difference--quotient composition identity expresses all $β=1,4$ blocks in terms of the projection kernel and bounded rational multipliers. From these formulas we give short steepest-descent proofs of bulk and edge universality (sine/Airy/Bessel) with uniform error control, an explicit Meixner$\to$Laguerre hard-edge crossover, and a first $A^{-1}$ correction that follows directly from the integrable structure. Finally, we show that Archimedean Kuznetsov tests splice into the Pfaffian kernels by a bounded holomorphic symbol acting in the contour variable; the symbol enters only through the same Cauchy difference--quotient, so the leading sine/Airy/Bessel limits persist and the $A^{-1}$ term again comes from linearizing at the saddle(s). |
| title | Integrable Contour Kernels in Discrete $β=1,4$ Ensembles, Universality and Kuznetsov Multipliers |
| topic | Mathematical Physics Number Theory Probability |
| url | https://arxiv.org/abs/2511.08356 |