The CS decomposition and conditional negative correlation inequalities for determinantal processes

TL;DR

This paper introduces new negative correlation inequalities for determinantal processes conditioned on certain events, utilizing geometric and algebraic tools like the CS decomposition and principal angles.

Contribution

It develops novel negative correlation inequalities for conditioned determinantal processes using geometric and algebraic methods.

Findings

Derived new inequalities for conditioned determinantal processes.

Utilized CS decomposition and principal angles for analysis.

Provided a geometric framework for understanding correlations.

Abstract

For a conditional process of the form $(\phi \vert A_{i} \not\subset \phi)$ where $\phi$ is a determinantal process we obtain a new negative correlation inequalities. Our approach relies upon the underlying geometric structure of the elementary discrete determinantal processes by using the canonical representation of a pair of subspaces in terms of principal vectors and angles, as well as the classical CS decomposition.