Difference operators and determinantal point processes

TL;DR

This paper explores how difference operators can be effectively used to analyze limit transitions in a family of determinantal point processes related to representation theory and random matrix theory, with processes on a one-dimensional lattice.

Contribution

It demonstrates the application of specific difference operators to study limit transitions within determinantal point processes on lattices, linking spectral projections to correlation kernels.

Findings

Difference operators correspond to spectral projections of selfadjoint operators.

Efficient methods for studying limit transitions in determinantal processes.

Connections between difference operators and correlation kernels.

Abstract

We consider a family {P} of determinantal point processes arising in representation theory and random matrix theory. The processes live on the one-dimensional lattice and their correlation kernels correspond to projection operators in the l^2 Hilbert space on the lattice. Moreover, these projections are spectral projections associated to certain selfadjoint second order difference operators on the lattice. The aim of the note is to demonstrate that the difference operators in question can be efficiently employed in the study of limit transitions inside the family {P}.