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Integrable Contour Kernels in Discrete $β=1,4$ Ensembles, Universality and Kuznetsov Multipliers

Miguel Tierz

TL;DR

The paper develops explicit double–contour (IIKS) representations for discrete β=1 and β=4 kernels associated with Meixner, Charlier, and Krawtchouk weights, unifying the orthogonal and symplectic cases through a single Cauchy–difference–quotient composition identity. This framework yields uniform proofs of bulk sine and edge Airy/Bessel universality, a Meixner→Laguerre hard–edge crossover, and an explicit A^{-1} correction arising from linearizing the difference–quotient at saddles. It further shows that archimedean Kuznetsov tests can be inserted as bounded contour multipliers without changing leading limits, while producing finite–size terms via the same linearization mechanism. The results connect integrable kernel theory with discrete random partitions and representation–theoretic models, offering practical tools for finite–N computations and potential crossovers to arithmetic deformations and number‑theoretic kernels. Overall, the work provides a versatile, uniform approach to universality and finite‑size corrections in discrete random matrix ensembles with broad implications for related combinatorial and number-theoretic structures.

Abstract

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).

Integrable Contour Kernels in Discrete $β=1,4$ Ensembles, Universality and Kuznetsov Multipliers

TL;DR

The paper develops explicit double–contour (IIKS) representations for discrete β=1 and β=4 kernels associated with Meixner, Charlier, and Krawtchouk weights, unifying the orthogonal and symplectic cases through a single Cauchy–difference–quotient composition identity. This framework yields uniform proofs of bulk sine and edge Airy/Bessel universality, a Meixner→Laguerre hard–edge crossover, and an explicit A^{-1} correction arising from linearizing the difference–quotient at saddles. It further shows that archimedean Kuznetsov tests can be inserted as bounded contour multipliers without changing leading limits, while producing finite–size terms via the same linearization mechanism. The results connect integrable kernel theory with discrete random partitions and representation–theoretic models, offering practical tools for finite–N computations and potential crossovers to arithmetic deformations and number‑theoretic kernels. Overall, the work provides a versatile, uniform approach to universality and finite‑size corrections in discrete random matrix ensembles with broad implications for related combinatorial and number-theoretic structures.

Abstract

We obtain explicit double-contour representations for the correlation kernels of the discrete orthogonal () and symplectic () random matrix ensembles with Meixner, Charlier, and Krawtchouk weights. A single Cauchy--difference--quotient composition identity expresses all 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 MeixnerLaguerre hard-edge crossover, and a first 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 term again comes from linearizing at the saddle(s).

Paper Structure

This paper contains 45 sections, 35 theorems, 133 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation
  3. Definition of discrete symplectic and orthogonal ensembles
  4. Meixner ensembles for $\beta=1,4$: integral formulas
  5. Meixner generating function and contour conventions
  6. The composition lemma and immediate corollary
  7. Meixner $\beta=1$ (orthogonal).
  8. Meixner $\beta=4$ (symplectic)
  9. Contours, poles and boundedness
  10. Common features emphasized
  11. The Charlier ensemble ($\beta=1,4$): integral formulas
  12. Single- and double-contour representations
  13. Discrete operators as rational multipliers in the $t$–plane
  14. Cauchy–multiplier composition in the $t$–plane
  15. Charlier $\beta=1$ (orthogonal): rank–one correction
  16. ...and 30 more sections

Key Result

Proposition 1.1

The operators $D$ and $\epsilon$ are mutual inverse (Christoffel–Darboux reductions used later can be found in Szego1975.) .

Figures (1)

  • Figure 1: Admissible contour configuration for the Kuznetsov multiplier. The branch cut is $(-\infty,0]$. The loops $\Gamma_1\subset\Gamma_2$ are positively oriented, lie in a slit sector $S_\delta$ (bounded away from the cut), and avoid $w=\pm1$.

Theorems & Definitions (71)

  • Proposition 1.1
  • Definition 1.2: Rational multipliers in the contour coordinates
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5: Meixner weight
  • Definition 2.6: Charlier (Poisson) weight
  • Definition 2.7: Krawtchouk weight
  • Lemma 3.1: Single–contour formulas and rational multipliers
  • ...and 61 more