Matrix kernels for measures on partitions

TL;DR

This paper derives explicit Pfaffian formulas and contour integral representations for correlation functions of z-measures on partitions with Jack parameters 2 or 1/2, linking representation theory and random matrix ensembles.

Contribution

It provides explicit Pfaffian kernel formulas for these measures and connects them to random matrix theory symmetries, advancing understanding of their correlation structures.

Findings

Correlation functions expressed as Pfaffians with matrix kernels

Explicit computation of these kernels

Contour integral representations for related measures

Abstract

We consider the problem of computation of the correlation functions for the z-measures with the deformation (Jack) parameters 2 or 1/2. Such measures on partitions are originated from the representation theory of the infinite symmetric group, and in many ways are similar to the ensembles of Random Matrix Theory of $\beta=4$ or $\beta=1$ symmetry types. For a certain class of such measures we show that correlation functions can be represented as Pfaffians including $2\times 2$ matrix valued kernels, and compute these kernels explicitly. We also give contour integral representations for correlation kernels of closely connected measures on partitions.