Random geometric subdivisions

TL;DR

This paper investigates various models of random geometric subdivisions, demonstrating convergence properties and limiting shapes such as parallelograms and flat triangles, extending prior work by Diaconis and Miclo.

Contribution

It introduces new models of geometric subdivisions and analyzes their asymptotic shapes and distributional convergence, expanding understanding of random geometric processes.

Findings

Limit shape of quadrilateral subdivision is a parallelogram.

Triangle subdivisions by angle bisectors converge weakly to a non-atomic distribution.

Random point subdivisions of triangles tend to a flat triangle.

Abstract

We study several models of random geometric subdivisions arising from the model of Diaconis and Miclo (2011). In particular, we show that the limiting shape of an indefinite subdivision of a quadrilateral is a.s.\ a parallelogram. We also show that the geometric subdivisions of a triangle by angle bisectors converge (only weakly) to a non-atomic distribution, and that the geometric subdivisions of a triangle by choosing random points on its sides converges to a "flat" triangle, similarly to the result of Diaconis and Miclo (2011).