Jante's law process

Abstract

Consider the process which starts with N ≥ 3 distinct points on R d , and fix a positive integer K < N . Of the total N points keep those N −K which minimize the energy (defined as the sum of all pairwise distances squared) amongst all the possible subsets of size N − K, and then replace the removed points by K i.i.d. points sampled according to some fixed distribution ζ. Repeat this process ad infinitum. We obtain various quite non-restrictive conditions under which the set of points converges to a certain limit. This is a very substantial generalization of the "Keynesian beauty contest process" introduced in [3], where K = 1 and the distribution ζ was uniform on the unit cube.