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Intersection of unit balls in classical matrix ensembles

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Abstract

We study the volume of the intersection of two unit balls from one of the classical matrix ensembles GOE, GUE and GSE, as the dimension tends to infinity. This can be regarded as a matrix analogue of a result of Schechtman and Schmuckenschläger for classical ℓp-balls [Schechtman and Schmuckenschläger, GAFA Lecture Notes, 1991]. The proof of our result is based on two ingredients, which are of independent interest. The first one is a weak law of large numbers for a point chosen uniformly at random in the unit ball of such a matrix ensemble. The second one is an explicit computation of the asymptotic volume of such matrix unit balls, which in turn is based on the theory of logarithmic potentials with external fields.

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Author notes

  1. Joscha Prochno

    Present address: Institute of Mathematics & Scientific Computing, Heinrichstraße 36, 8010, Graz, Austria

Authors and Affiliations

  1. Institut für Mathematische Stochastik, Westfälische Wilhelms-Universität Münster, Orléans-Ring 10, 48149, Münster, Germany

    Zakhar Kabluchko

  2. School of Mathematics & Physical Sciences, University of Hull, Hull, HU6 7RX, United Kingdom

    Joscha Prochno

  3. Fakultät für Mathematik, Ruhr-Universität Bochum, Bochum, 44780, Germany

    Christoph Thäle

Authors
  1. Zakhar Kabluchko
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  2. Joscha Prochno
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Correspondence to Christoph Thäle.

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Kabluchko, Z., Prochno, J. & Thäle, C. Intersection of unit balls in classical matrix ensembles. Isr. J. Math. 239, 129–172 (2020). https://doi.org/10.1007/s11856-020-2052-6

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  • DOI: https://doi.org/10.1007/s11856-020-2052-6

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