Electronic Journal of Probability

Are random permutations spherically uniform?

Michael D. Perlman

Full-text: Open access

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

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1580267009

Digital Object Identifier
doi:10.1214/20-EJP418

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

Keywords
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

Rights
Creative Commons Attribution 4.0 International License.

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


Export citation

References

  • [1] Alishahi, K. and Zamani, M. S. (2015). The spherical ensemble and uniform distribution of points on the sphere. Electronic J. Prob. 20 1–27.
  • [2] Baek, J. and Adams, A. (2009). Some useful properties of the permutohedral lattice for Gaussian filtering. http://graphics.stanford.edu/papers/permutohedral/permutohedral_techreport.pdf.
  • [3] Ball, K. and Perissinaki, I. (1998). The subindependence of coordinate slabs in $\ell _{p}^{n}$ balls. Israel J. Math. 107 289–299.
  • [4] Barthe, F., Gamboa, F., Lozada-Chang, L., Rouault, A. (2010). Generalized Dirichlet distributions on the ball and moments. Latin American J. Prob. Math. Statist. 7 319–340.
  • [5] Beck, J. (1984). Sums of distances between points on a sphere – an application of the theoruy of irregularities of distribution to discrete geometry. Mathematika 31 33–41.
  • [6] Brauchart, J. S. and Grabner, P. J. (2015). Distributing many points on spheres: minimal energy and designs. J. Complexity 311 293–326.
  • [7] Das Gupta, S., Eaton, M. L., Olkin, I., Perlman, M. D., Savage, L. J., Sobel, M. (1972). Inequalities for the probability content of convex regions for elliptically contoured distributions. Proc. Sixth Berkely Symp. Math. Statist. Prob. 2 241–265.
  • [8] Diaconis, P. and Freedman, D. (1987). A dozen de Finetti-style results in search of a theory. Ann. Inst. H. Poincare Probab. Statist. 23 397–423.
  • [9] Eaton, M. L. (1989). Group Invariance Applications in Statistics. Regional Conference Series in Probability and Statistics Vol. 1, Institute of Mathematical Statistics.
  • [10] Eaton, M. L. and Perlman, M. D. (1977). Reflection groups, generalized Schur functions, and the geometry of majorization. Ann. Prob. 5 829–860.
  • [11] Fung, T. and Seneta, E. (2018). Quantile function expansion using regularly varying functions. Method. Comp. Appl. Probab. 20 1091–1103.
  • [12] Grove, L. C. and Benson, C. T. (1985). Finite Reflection Groups, 2nd Ed. Springer, New York.
  • [13] Leopardi, P. (2007). Distributing points on the sphere: partitions, separation, quadrature and energy. Ph.D. thesis, The University of New South Wales (2007).
  • [14] Leopardi, P. (2013). Discrepancy, separation and Riesz energy of finite point sets on the unit sphere. Adv. Comp. Math. 39 27–43.
  • [15] Marshall, A. W. and Olkin, I. (1979). Inequalities: Theory of Majorization and Its Applications. Academic Press, New York.
  • [16] Qi, F. and Guo, B.-N. (2011). Sharp bounds for harmonic numbers. Applied Math. and Computation 218 991–995.
  • [17] Schmidt, W. M. (1969). Irregularities of distribution IV. Inventiones Math. 78 55–82.
  • [18] Stewart, K. (2018). Total variation approximation of random orthogonal matrices by Gaussian matrices. J. Theoretical Probab. https://doi.org/10.1007/s10959-019-00900-5.
  • [19] Stolarsky, K. B. (1973). Sums of distances between points on a sphere II. Proc. Amer. Math. Soc. 41 575–582.
  • [20] Wendel, J. G. (1948). Note on the gamma function. Amer. Math. Monthly 55 563–564.