Scale invariant growth processes in expanding space

TL;DR

This paper establishes an exact relation linking the statistical properties of growth structures in expanding and fixed geometries, generalizing conformal transformations and aiding the understanding of complex fractal patterns.

Contribution

It introduces a novel, exact relation that preserves local scale invariance in growth processes across expanding and fixed spaces, extending conformal symmetry concepts.

Findings

Derived an exact relation between expanding and fixed geometries.

Numerically demonstrated the relation for coalescing Lévy flights and fractional Brownian motions.

Provided explicit asymptotic statistics for growth in expanding domains.

Abstract

Many growth processes lead to intriguing stochastic patterns and complex fractal structures which exhibit local scale invariance properties. Such structures can often be described effectively by space-time trajectories of interacting particles, and their large scale behaviour depends on the overall growth geometry. We establish an exact relation between statistical properties of structures in uniformly expanding and fixed geometries, which preserves the local scale invariance and is independent of other properties such as the dimensionality. This relation generalizes standard conformal transformations as the natural symmetry of self-affine growth processes. We illustrate our main result numerically for various structures of coalescing L\'evy flights and fractional Brownian motions, including also branching and finite particle sizes. One of the main benefits of this new approach is a full, explicit description of the asymptotic statistics in expanding domains, which are often non-trivial and random due to amplification of initial fluctuations.