Hostname: page-component-8448b6f56d-tj2md Total loading time: 0 Render date: 2024-04-24T08:35:38.827Z Has data issue: false hasContentIssue false
  • English
  • Français

On Barycentric Subdivision

Published online by Cambridge University Press:  24 November 2010

PERSI DIACONIS  and
LAURENT MICLO
PERSI DIACONIS
Affiliation:
Departments of Statistics and Mathematics, Stanford University, USA (e-mail: diaconis@math.stanford.edu)
LAURENT MICLO
Affiliation:
Institut de Mathématiques de Toulouse, UMR 5219, Université de Toulouse and CNRS, France (e-mail: miclo@math.univ-toulouse.fr)
Get access Rights & Permissions [Opens in a new window]

Abstract

Consider the barycentric subdivision which cuts a given triangle along its medians to produce six new triangles. Uniformly choosing one of them and iterating this procedure gives rise to a Markov chain. We show that, almost surely, the triangles forming this chain become flatter and flatter in the sense that their isoperimetric values go to infinity with time. Nevertheless, if the triangles are renormalized through a similitude to have their longest edge equal to [0, 1] ⊂ ℂ (with 0 also adjacent to the shortest edge), their aspect does not converge and we identify the limit set of the opposite vertex with the segment [0, 1/2]. In addition we prove that the largest angle converges to π in probability. Our approach is probabilistic, and these results are deduced from the investigation of a limit iterated random function Markov chain living on the segment [0, 1/2]. The stationary distribution of this limit chain is particularly important in our study.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 20 , Issue 2 , March 2011 , pp. 213 - 237
Copyright
Copyright © Cambridge University Press 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Bárány, I., Beardon, A. F. and Carne, T. K. (1996) Barycentric subdivision of triangles and semigroups of Möbius maps. Mathematika 43 165–171.CrossRefGoogle Scholar
[2]Barnsley, M. F. and Elton, J. H. (1988) A new class of Markov processes for image encoding. Adv. Appl. Probab. 20 14–32.CrossRefGoogle Scholar
[3]Blackwell, D. (2008) Barycentric subdivision. Private communication.Google Scholar
[4]Butler, S. and Graham, R. (2010) Iterated triangle partitions. Preprint available at: http://www.math.ucla.edu/~butler/PDF/iterated_triangles.pdf.CrossRefGoogle Scholar
[5]Dellacherie, C. and Meyer, P.-A. (1980) Probabilités et Potentiel, Chap. V–VIII, Vol. 1385 of Actualités Scientifiques et Industrielles (Current Scientific and Industrial Topics), revised edition, Hermann.Google Scholar
[6]Diaconis, P. and Freedman, D. (1999) Iterated random functions. SIAM Rev. 41 45–76.CrossRefGoogle Scholar
[7]Diaconis, P. and McMullen, C. T. (2010) Barycentric subdivision. Article in preparation.CrossRefGoogle Scholar
[8]Diaconis, P. and Miclo, L. (2010) On barycentric partitions, with simulations. Preprint available at: http://hal.archives-ouvertes.fr and http://arxiv.org/.CrossRefGoogle Scholar
[9]Dubins, L. E. and Freedman, D. A. (1966) Invariant probabilities for certain Markov processes. Ann. Math. Statist. 37 837–848.CrossRefGoogle Scholar
[10]Hough, B. (2009) Tessellation of a triangle by repeated barycentric subdivision. Electron. Commun. Probab. 14 270–277.CrossRefGoogle Scholar
[11]Osserman, R. (1978) The isoperimetric inequality. Bull. Amer. Math. Soc. 84 1182–1238.CrossRefGoogle Scholar
[12]Stout, W. F. (1974) Almost Sure Convergence, Vol. 24 of Probability and Mathematical Statistics, Academic Press.Google Scholar