Intersection of unit balls in classical matrix ensembles
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INTERSECTION OF UNIT BALLS IN CLASSICAL MATRIX ENSEMBLES BY
Zakhar Kabluchko Institut f¨ ur Mathematische Stochastik, Westf¨ alische Wilhelms-Universit¨ at M¨ unster Orl´eans-Ring 10, 48149 M¨ unster, Germany e-mail: [email protected] AND
Joscha Prochno∗ School of Mathematics & Physical Sciences, University of Hull Hull HU6 7RX, United Kingdom e-mail: [email protected] AND
¨le Christoph Tha Fakult¨ at f¨ ur Mathematik, Ruhr-Universit¨ at Bochum, Bochum 44780, Germany e-mail: [email protected]
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 Schmuckenschl¨ ager for classical p -balls [Schechtman and Schmuckenschl¨ ager, 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.
∗ Current address: Institute of Mathematics & Scientific Computing, Heinrich-
straße 36, 8010 Graz, Austria. Received June 23, 2018 and in revised form September 2, 2019
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¨ Z. KABLUCHKO, J. PROCHNO AND C. THALE
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Isr. J. Math.
Contents
1.
Introduction and main result . . . . . . . . . . . . . . . .
2
2.
Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . .
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3.
Asymptotic volume of matrix balls . . . . . . . . . . . . .
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4.
Random sampling in matrix balls and a weak law of large numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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5.
Application to high-dimensional intersections . . . . . . .
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References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
1. Introduction and main result To understand the geometry of high-dimensional convex bodies and, in particular, the distribution of volume is one of the central aspects considered in Asymptotic Geometric Analysis. It has been realized by now that such an understanding has important connections and implications to various questions considered in other branches of mathematics and related disciplines. We refer the reader to the research monographs and surveys [2, 3, 8, 9] for background information. Ever since, there has been a particular interest and focus on the non-commutative setting of Schatten trace classes or classical matrix ensembles as is demonstrated by the research carried out in [4, 10, 12, 15, 19, 26, 34]. While this often underlines a similarity to the commutative setting of classical p sequence spaces, it also shows differences in the behavior of certain quantities related to the geometry of Banach spaces. In fact, often different methods and tools are needed and proofs can be considerably more involved. In the classical setting of np -balls, Schechtm
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