A universal result for consecutive random subdivision of polygons

Tuan-Minh Nguyen, Stanislav Volkov

Research output: Contribution to journalArticleResearchpeer-review

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 [11] which are achieved mostly by using ad hoc methods.

Original languageEnglish
Pages (from-to)341-371
Number of pages31
JournalRandom Structures & Algorithms
Volume51
Issue number2
DOIs
Publication statusPublished - 1 Sep 2017
Externally publishedYes

Keywords

  • Lyapunov exponents
  • products of random matrices
  • random subdivisions

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