A universal result for consecutive random subdivision of polygons
(2017) In Random Structures & Algorithms 51(2). p.341371 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 nondegenerateness 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 nondegenerateness 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:
http://lup.lub.lu.se/record/1015ded431f74c4993e4a11a8b26d0e0
 author
 Nguyen, Tuan Minh ^{LU} and Volkov, Stanislav ^{LU}
 organization
 publishing date
 20170721
 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
 10982418
 DOI
 10.1002/rsa.20702
 language
 English
 LU publication?
 yes
 id
 1015ded431f74c4993e4a11a8b26d0e0
 date added to LUP
 20170303 14:01:26
 date last changed
 20180107 11:53:57
@article{1015ded431f74c4993e4a11a8b26d0e0, 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 nondegenerateness 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 = {10982418}, keyword = {random subdivisions,products of random matrices,Lyapunov exponents}, language = {eng}, month = {07}, number = {2}, pages = {341371}, 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}, }