# Jante's law process

## TL;DR

This paper studies a dynamic process where points in Euclidean space are iteratively selected based on minimal energy configurations, replaced by random points, and under certain conditions, the points converge to a limit, generalizing previous models.

## Contribution

It introduces a generalized Jante's law process with arbitrary K and distribution or the replaced points, extending prior work on the Keynesian beauty contest process.

## Key findings

- Points converge to a limit under broad conditions
- Generalization of previous models with arbitrary K and distributions
- Provides conditions for convergence of the process

## Abstract

Consider the process which starts with $N\ge 3$ distinct points on ${\mathbb R}^d$, 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 $\zeta$. 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" studied by Grinfeld, Volkov and Wade, where $K=1$ and the distribution $\zeta$ was uniform on the unit cube.