Mean-field approach to Random Apollonian Packing

TL;DR

This paper develops a mean-field model to analyze the scaling properties and fractal dimensions of Random Apollonian Packing in 2, 3, and 4 dimensions, validated by extensive simulations.

Contribution

It introduces a novel mean-field approach that accurately predicts the insertion probability and scaling behavior without assuming a predefined radius distribution.

Findings

Insertion probability matches numerical simulations across dimensions.

The model accurately predicts fractal dimensions of the packing.

Validation with large-scale simulations confirms the model's effectiveness.

Abstract

We revisit the scaling properties of growing spheres randomly seeded in d=2,3 and 4 dimensions using a mean-field approach. We model the insertion probability without assuming a priori a functional form for the radius distribution. The functional form of the insertion probability shows an unprecedented agreement with numerical simulations in d=2, 3 and 4 dimensions. We infer from the insertion probability the scaling behavior of the Random Apollonian Packing and its fractal dimensions. The validity of our model is assessed with sets of 256 simulations each containing 20 million spheres in 2, 3 and 4 dimensions.