Hyperdeterminantal point processes

TL;DR

This paper introduces hyperdeterminantal point processes, a novel generalization of determinantal point processes involving interactions among 2M points, extending the mathematical framework using hyperdeterminants.

Contribution

It initiates the study of hyperdeterminantal point processes, generalizing determinantal processes with interactions among multiple points using hyperdeterminants.

Findings

Some properties of determinantal point processes are preserved in the hyperdeterminantal generalization.

The framework involves hypercubic arrays and Cayley's hyperdeterminant.

Potential applications in modeling complex interactions among points.

Abstract

As well as arising naturally in the study of non-intersecting random paths, random spanning trees, and eigenvalues of random matrices, determinantal point processes (sometimes also called fermionic point processes) are relatively easy to simulate and provide a quite broad class of models that exhibit repulsion between points. The fundamental ingredient used to construct a determinantal point process is a kernel giving the pairwise interactions between points: the joint distribution of any number of points then has a simple expression in terms of determinants of certain matrices defined from this kernel. In this paper we initiate the study of an analogous class of point processes that are defined in terms of a kernel giving the interaction between $2M$ points for some integer $M$. The role of matrices is now played by $2M$-dimensional "hypercubic" arrays, and the determinant is replaced by a suitable generalization of it to such arrays -- Cayley's first hyperdeterminant. We show that some of the desirable features of determinantal point processes continue to be exhibited by this generalization.