A universal result for consecutive random subdivision of polygons

TL;DR

This paper studies the long-term shape evolution of polygons undergoing repeated random subdivisions, showing they tend to become flat, with explicit convergence rates for triangles linked to Lyapunov exponents, generalizing previous results.

Contribution

It provides a general framework using random matrix theory to analyze polygon flattening, extending prior ad hoc methods and calculating exact convergence rates for triangles.

Findings

Polygons tend to become flat after repeated subdivisions.

Convergence rate for triangles is explicitly calculated.

Results are based on products of random matrices and Lyapunov exponents.

Abstract

We consider consecutive random subdivision of polygons described as follows. Given an initial convex polygon with $d\ge 3$ 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 $\xi$. 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 $\xi$, 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.