Are random permutations spherically uniform?

For large q, does the (discrete) uniform distribution on the set of q! permutations of the vector x̄ = (1, 2, . . . , 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, . . . , q) is not the most favorable configuration for spherical uniformity of permutations. A more favorable configuration x̂ is found, namely that which minimizes the normalized surface area of the largest empty spherical cap among its q! permutations. Unlike that for x̄, the normalized surface area of the largest empty spherical cap among the permutations of x̂ approaches 0 as q →∞. Nonetheless the permutations of x̂ do not approach spherical uniformity either. The existence of an asymptotically spherically uniform permutation sequence remains an open question.

• Publication year

  2020

• Venue

  Electronic Journal of Probability

• Publication date

  Unknown publication date

• Fields of study

  Mathematics

• Identifiers
  DOI 10.1214/20-ejp418
• External record

  Open on Semantic Scholar

• Source metadata

  Semantic Scholar

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• No concepts are published for this paper.

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