Determinantal point processes from symplectic and orthogonal characters and applications

TL;DR

This paper demonstrates that symplectic and orthogonal character analogues of certain measures are determinantal, enabling new results on Toeplitz+Hankel determinants, limit theorems, and asymptotic behaviors in partition measures.

Contribution

It introduces determinantal structures for symplectic and orthogonal characters, extending known results and deriving new asymptotic and limit theorems in partition measures.

Findings

Determinantal formulas for symplectic and orthogonal character measures

Szegő-type limit theorem established

Asymptotic results for symplectic and orthogonal analogues of Poissonized Plancherel measure

Abstract

We show that the symplectic and orthogonal character analogues of Okounkov's Schur measure (on integer partitions) are determinantal, with explicit correlation kernels. We apply this to prove certain Borodin-Okounkov-Gessel-type results concerning Toeplitz+Hankel and Fredholm determinants; a Szeg\H{o}-type limit theorem; an edge Baik-Deift-Johansson-type asymptotical result for certain symplectic and orthogonal analogues of the poissonized Plancherel measure; and a similar result for actual poissonized Plancherel measures supported on "almost symmetric" partitions.