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A universal result for consecutive random subdivision of polygons

Nguyen, Tuan Minh LU and Volkov, Stanislav LU (2017) In Random Structures & Algorithms 51(2). p.341-371
Abstract
We consider consecutive random subdivision of polygons described as follows. Given an initial convex polygon with d≥3 edges, we choose a point at random on each edge, such that the proportions in which these points divide edges are i.i.d. copies of some random variable ξ. These new points form a new (smaller) polygon. By repeatedly implementing this procedure we obtain a sequence of random polygons. The aim of this paper is to show that under very mild non-degenerateness conditions on ξ, the shapes of these polygons eventually become "flat" The convergence rate to flatness is also investigated; in particular, in the case of triangles (d=3), we show how to calculate the exact value of the rate of convergence, connected to Lyapunov... (More)
We consider consecutive random subdivision of polygons described as follows. Given an initial convex polygon with d≥3 edges, we choose a point at random on each edge, such that the proportions in which these points divide edges are i.i.d. copies of some random variable ξ. These new points form a new (smaller) polygon. By repeatedly implementing this procedure we obtain a sequence of random polygons. The aim of this paper is to show that under very mild non-degenerateness conditions on ξ, the shapes of these polygons eventually become "flat" The convergence rate to flatness is also investigated; in particular, in the case of triangles (d=3), we show how to calculate the exact value of the rate of convergence, connected to Lyapunov exponents. Using the theory of products of random matrices our paper greatly generalizes the results of Volkov (2013) which are achieved mostly by using ad hoc methods. (Less)
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author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
random subdivisions, products of random matrices, Lyapunov exponents
in
Random Structures & Algorithms
volume
51
issue
2
pages
31 pages
publisher
John Wiley & Sons
external identifiers
  • scopus:85025141310
  • wos:000406861100006
ISSN
1098-2418
DOI
10.1002/rsa.20702
language
English
LU publication?
yes
id
1015ded4-31f7-4c49-93e4-a11a8b26d0e0
date added to LUP
2017-03-03 14:01:26
date last changed
2018-01-07 11:53:57
@article{1015ded4-31f7-4c49-93e4-a11a8b26d0e0,
  abstract     = {We consider consecutive random subdivision of polygons described as follows. Given an initial convex polygon with d≥3 edges, we choose a point at random on each edge, such that the proportions in which these points divide edges are i.i.d. copies of some random variable ξ. These new points form a new (smaller) polygon. By repeatedly implementing this procedure we obtain a sequence of random polygons. The aim of this paper is to show that under very mild non-degenerateness conditions on ξ, the shapes of these polygons eventually become "flat" The convergence rate to flatness is also investigated; in particular, in the case of triangles (d=3), we show how to calculate the exact value of the rate of convergence, connected to Lyapunov exponents. Using the theory of products of random matrices our paper greatly generalizes the results of Volkov (2013) which are achieved mostly by using ad hoc methods.},
  author       = {Nguyen, Tuan Minh and Volkov, Stanislav},
  issn         = {1098-2418},
  keyword      = {random subdivisions,products of random matrices,Lyapunov exponents},
  language     = {eng},
  month        = {07},
  number       = {2},
  pages        = {341--371},
  publisher    = {John Wiley & Sons},
  series       = {Random Structures & Algorithms},
  title        = {A universal result for consecutive random subdivision of polygons},
  url          = {http://dx.doi.org/10.1002/rsa.20702},
  volume       = {51},
  year         = {2017},
}