Extremal statistics on non-crossing configurations

We analyze extremal statistics in non-crossing configurations on the n vertices of a convex polygon. We prove that the maximum degree and the largest component are of logarithmic order, and that, suitably scaled, they converge to a well-defined constant. We also prove that the diameter is of order n . The proofs are based on singularity analysis, an application of the first and second moment method, and on the analysis of iterated functions.