## Electronic Journal of Probability

### Are random permutations spherically uniform?

Michael D. Perlman

#### 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.

#### Article information

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

Dates
Accepted: 13 January 2020
First available in Project Euclid: 29 January 2020

https://projecteuclid.org/euclid.ejp/1580267009

Digital Object Identifier
doi:10.1214/20-EJP418

Subjects
Secondary: 60C05: Combinatorial probability

#### Citation

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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