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2020 Are random permutations spherically uniform?
Michael D. Perlman
Electron. J. Probab. 25(none): 1-26 (2020). DOI: 10.1214/20-EJP418

Abstract

For large $q$, does the (discrete) uniform distribution on the set of $q!$ permutations of the vector $\bar{\mathbf {x}} ^{q}=(1,2,\dots ,q)'$ closely approximate the (continuous) uniform distribution on the $(q-2)$-sphere that contains them? These permutations comprise the vertices of the regular permutohedron, a $(q-1)$-dimensional convex polyhedron. The answer is emphatically no: these permutations are confined to a negligible portion of the sphere, and the regular permutohedron occupies a negligible portion of the ball. However, $(1,2,\dots ,q)$ is not the most favorable configuration for spherical uniformity of permutations. A more favorable configuration $\hat{\mathbf {x}} ^{q}$ is found, namely that which minimizes the normalized surface area of the largest empty spherical cap among its $q!$ permutations. Unlike that for $\bar{\mathbf {x}} ^{q}$, the normalized surface area of the largest empty spherical cap among the permutations of $\hat{\mathbf {x}} ^{q}$ approaches $0$ as $q\to \infty $. Nonetheless the permutations of $\hat{\mathbf {x}} ^{q}$ do not approach spherical uniformity either. The existence of an asymptotically spherically uniform permutation sequence remains an open question.

Citation

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Michael D. Perlman. "Are random permutations spherically uniform?." Electron. J. Probab. 25 1 - 26, 2020. https://doi.org/10.1214/20-EJP418

Information

Received: 5 March 2019; Accepted: 13 January 2020; Published: 2020
First available in Project Euclid: 29 January 2020

zbMATH: 1446.11145
MathSciNet: MR4059187
Digital Object Identifier: 10.1214/20-EJP418

Subjects:
Primary: 11K38
Secondary: 60C05

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