A determinantal point process governed by an integrable projection kernel is Giambelli compatible

TL;DR

This paper proves new determinantal identities for the expectations of scaled characteristic polynomials in certain determinantal point processes, extending classical results and establishing stability of the Giambelli formula under averaging.

Contribution

It introduces determinantal identities for scaled characteristic polynomial expectations and proves the stability of the Giambelli formula in integrable projection kernel processes.

Findings

Established determinantal identities (7) and (8) for scaled characteristic polynomial expectations.

Proved the stability of the Giambelli formula under averaging for these processes.

Characterized conditional measures as orthogonal polynomial ensembles.

Abstract

The first main result of this note, Theorem 1.2, establishes the determinantal identities (7) and (8) for the expectation, under a determinantal point process governed by an integrable projection kernel, of scaling limits of characteristic polynomials sampled at several points. The determinantal identities (7) and (8) can be seen as the scaling limit of the identity of Fyodorov and Strahov for the averages of ratios of products of the values of the characteristic polynomial of a Gaussian unitary matrix. Borodin, Olshanski and Strahov derived the determinantal identity of Fyodorov and Strahov from the stability of the Giambelli formula under averaging. In Theorem 1.4 the stability of the Giambelli formula under averaging is established for determinantal point process with integrable projection kernels. The proof of Theorems 1.2 and 1.4 relies on the characterization of conditional measures of our point processes as orthogonal polynomial ensembles.