Convergence to collinearity of a sequence of random triangle shapes

Abstract

A sequence of random triangles is constructed by choosing successively the three vertices of one triangle at random in the interior of its predecessor. A way is found to prove that the shapes of these triangles converge, almost surely, to collinear shapes, thus closing a gap in one of the central arguments of Mannion [5]. The new approach is based on a representation of the triangle process by a sequence of products of i.i.d. random matrices. We succeed in calculating the corresponding Lyapounov exponent.