Scale-invariant growth processes in expanding space

Ali, A. and Grosskinsky, S. and Ball, R. C. and Somfai, Ellák (2013) Scale-invariant growth processes in expanding space. PHYSICAL REVIEW E - STATISTICAL, NONLINEAR AND SOFT MATTER PHYSICS (2001-2015), 87 (2). ISSN 1539-3755

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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 inde- pendent 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 re- sult 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.

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