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)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/1015ded4-31f7-4c49-93e4-a11a8b26d0e0
- author
- Nguyen, Tuan Minh
LU
and Volkov, Stanislav
LU
- organization
- publishing date
- 2017-07-21
- 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 Inc.
- 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
- 2022-03-09 01:28:00
@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}},
keywords = {{random subdivisions; products of random matrices; Lyapunov exponents}},
language = {{eng}},
month = {{07}},
number = {{2}},
pages = {{341--371}},
publisher = {{John Wiley & Sons Inc.}},
series = {{Random Structures & Algorithms}},
title = {{A universal result for consecutive random subdivision of polygons}},
url = {{http://dx.doi.org/10.1002/rsa.20702}},
doi = {{10.1002/rsa.20702}},
volume = {{51}},
year = {{2017}},
}