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

TL;DR

This paper derives explicit formulas for correlation kernels in discrete $eta=1,4$ ensembles, proving universality and crossover phenomena, and connecting Kuznetsov multipliers with Pfaffian kernels.

Contribution

It provides explicit double-contour formulas for $eta=1,4$ kernels, establishes universality with error bounds, and links Kuznetsov multipliers to Pfaffian kernels in these ensembles.

Findings

Explicit double-contour kernel representations for $eta=1,4$ ensembles.

Proof of bulk and edge universality with uniform error control.

Identification of the role of Kuznetsov multipliers in Pfaffian kernels.

Abstract

We obtain explicit double-contour representations for the correlation kernels of the discrete orthogonal ($\beta=1$) and symplectic ($\beta=4$) random matrix ensembles with Meixner, Charlier, and Krawtchouk weights. A single Cauchy--difference--quotient composition identity expresses all $\beta=1,4$ 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).