Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Any orthonormal basis in high dimension is uniformly distributed over the sphere

Sheldon Goldstein, Joel L. Lebowitz, Roderich Tumulka, and Nino Zanghî

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Abstract

Let $\mathbb{X}^{d}$ be a real or complex Hilbert space of finite but large dimension $d$, let $\mathbb{S}(\mathbb{X}^{d})$ denote the unit sphere of $\mathbb{X}^{d}$, and let $u$ denote the normalized uniform measure on $\mathbb{S}(\mathbb{X}^{d})$. For a finite subset $B$ of $\mathbb{S}(\mathbb{X}^{d})$, we may test whether it is approximately uniformly distributed over the sphere by choosing a partition $A_{1},\ldots,A_{m}$ of $\mathbb{S}(\mathbb{X}^{d})$ and checking whether the fraction of points in $B$ that lie in $A_{k}$ is close to $u(A_{k})$ for each $k=1,\ldots,m$. We show that if $B$ is any orthonormal basis of $\mathbb{X}^{d}$ and $m$ is not too large, then, if we randomize the test by applying a random rotation to the sets $A_{1},\ldots,A_{m}$, $B$ will pass the random test with probability close to 1. This statement is related to, but not entailed by, the law of large numbers. An application of this fact in quantum statistical mechanics is briefly described.

Résumé

Soit $\mathbb{X}^{d}$ un espace de Hilbert réel ou complexe de dimension finie mais grande $d$ et soit $\mathbb{S}(\mathbb{X}^{d})$ la sphère unité de $\mathbb{X}^{d}$, on note $u$ la mesure uniforme normalisée sur $\mathbb{S}(\mathbb{X}^{d})$. Pour un sous ensemble fini $B$ de $\mathbb{S}(\mathbb{X}^{d})$, nous pouvons tester s’il est approximativement uniformément distribué sur la sphère en choisissant une partition $A_{1},\ldots,A_{m}$ de $\mathbb{S}(\mathbb{X}^{d})$ et en vérifiant si la fraction des points dans $B$ qui se trouvent dans $A_{k}$ est proche de $u(A_{k})$ pour tout $k=1,\ldots,m$. Nous montrons que si $B$ est n’importe quelle base orthonormée de $\mathbb{X}^{d}$ et que si $m$ n’est pas trop grand, alors si on randomise le test en appliquant une rotation aléatoire aux ensembles $A_{1},\ldots,A_{m}$, l’ensemble $B$ va passer le test avec une probabilité proche de 1. Ce résultat est relié à la loi des grands nombres. Une application de ce résultat en mécanique statistique quantique est décrite brièvement.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist. Volume 53, Number 2 (2017), 701-717.

Dates
Received: 11 February 2015
Revised: 5 November 2015
Accepted: 15 November 2015
First available in Project Euclid: 11 April 2017

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1491897742

Digital Object Identifier
doi:10.1214/15-AIHP732

Subjects
Primary: 60F05: Central limit and other weak theorems 82B10: Quantum equilibrium statistical mechanics (general) 28C10: Set functions and measures on topological groups or semigroups, Haar measures, invariant measures [See also 22Axx, 43A05]

Keywords
Law of large numbers Haar measure on the orthogonal or unitary groups Asymptotics in high dimension Irreducible representations of the orthogonal or unitary groups random orthonormal basis

Citation

Goldstein, Sheldon; Lebowitz, Joel L.; Tumulka, Roderich; Zanghî, Nino. Any orthonormal basis in high dimension is uniformly distributed over the sphere. Ann. Inst. H. Poincaré Probab. Statist. 53 (2017), no. 2, 701--717. doi:10.1214/15-AIHP732. https://projecteuclid.org/euclid.aihp/1491897742.


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