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Hyperdeterminantal point processes

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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.

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Authors and Affiliations

  1. Department of Statistics #3860, University of California at Berkeley, 367 Evans Hall, Berkeley, CA, 94720-3860, USA

    Steven N. Evans

  2. Faculty of Mathematics, Wolfgang Pauli Institute, UZA 4 (7th floor, Green Area “C”), Nordbergstrasse 15, 1090, Vienna, Austria

    Alex Gottlieb

Authors
  1. Steven N. Evans
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  2. Alex Gottlieb
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Corresponding author

Correspondence to Steven N. Evans.

Additional information

S. N. Evans supported in part by NSF grant DMS-0405778. A. Gottlieb supported by the Vienna Science and Technology Fund, via the project “Correlation in quantum systems”.

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Evans, S.N., Gottlieb, A. Hyperdeterminantal point processes. Metrika 69, 85–99 (2009). https://doi.org/10.1007/s00184-008-0209-0

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  • DOI: https://doi.org/10.1007/s00184-008-0209-0

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