A universal result for consecutive random subdivision of polygons. Issue 2 (4th November 2016)

• Record Type: Journal Article
• Authors: Nguyen, Tuan‐Minh; Volkov, Stanislav
• Is Part Of: Random structures & algorithms. Volume 51:Issue 2(2017)
• Page Start: 341
• Page End: 371
• Publication Date: 2016-11-04
• Subjects: random subdivisions -- products of random matrices -- Lyapunov exponents; Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20702
• Languages: English
• ISSNs: 1042-9832

Abstract

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. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 51, 341–371, 2017