# Equivalent symmetric kernels of determinantal point processes

## TL;DR

This paper classifies all transformations of symmetric kernels in determinantal point processes that leave their correlation functions unchanged, providing a comprehensive understanding of kernel equivalences.

## Contribution

It provides a complete classification of kernel transformations that preserve correlation functions in symmetric determinantal point processes.

## Key findings

- Identifies all kernel transformations preserving correlation functions.
- Establishes conditions for kernel equivalence in symmetric determinantal processes.
- Enhances understanding of kernel non-uniqueness in determinantal point processes.

## Abstract

Determinantal point processes are point processes whose correlation functions are given by determinants of matrices. The entries of these matrices are given by one fixed function of two variables, which is called the kernel of the point process. It is well-known that there are different kernels that induce the same correlation functions. We classify all the possible transformations of a kernel that leave the induced correlation functions invariant, restricting to the case of symmetric kernels.