Electronic Journal of Probability

Are random permutations spherically uniform?

Michael D. Perlman

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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.

Article information

Electron. J. Probab., Volume 25 (2020), paper no. 9, 26 pp.

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

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Digital Object Identifier

Primary: 11K38: Irregularities of distribution, discrepancy [See also 11Nxx]
Secondary: 60C05: Combinatorial probability

permutations uniform distribution spherical cap discrepancy largest empty cap regular configuration regular permutohedron $L$-minimal configuration $L$-minimal permutohedron normal configuration normal permutohedron majorization spherical code permutation code

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Perlman, Michael D. Are random permutations spherically uniform?. Electron. J. Probab. 25 (2020), paper no. 9, 26 pp. doi:10.1214/20-EJP418. https://projecteuclid.org/euclid.ejp/1580267009

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