Random geometric subdivisions

We study several models of random geometric subdivisions arising from the model of Diaconis and Miclo (Combin Probab Comput 20 (2011) 213-237). 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 (Combin Probab Comput 20 (2011) 213-237). (c) 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013