Extremal statistics on non-crossing configurations

We analyze extremal statistics in non-crossing configurations on the nn 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.