From radial symmetry to fractal behavior of aggregation equilibria for repulsive-attractive potentials

TL;DR

This paper establishes conditions under which local minimizers of interaction energy with repulsive-attractive potentials are radially symmetric and unique, and introduces a new concept of potential concavity revealing fractal-like structures in minimizers.

Contribution

It provides generic conditions for symmetry and uniqueness of local minimizers and introduces a novel notion of potential concavity leading to fractal behaviors in aggregation models.

Findings

Radial symmetry of local minimizers under certain conditions

Uniqueness of local minimizers in the Wasserstein topology

Existence of fractal-like structures in minimizers

Abstract

For the interaction energy with repulsive-attractive potentials, we give generic conditions which guarantee the radial symmetry of the local minimizers in the infinite Wasserstein distance. As a consequence, we obtain the uniqueness of local minimizers in this topology for a class of interaction potentials. We introduce a novel notion of concavity of the interaction potential allowing us to show certain fractal-like behavior of the local minimizers. We provide a family of interaction potentials such that the support of the associated local minimizers has no isolated points and any superlevel set has no interior points.