Scale-invariant growth processes in expanding space

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 behavior 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 Lévy flights and fractional Brownian motions, including also branching and finite particle sizes. One of the main benefits of this approach is a full, explicit description of the asymptotic statistics in expanding domains, which are often nontrivial and random due to amplification of initial fluctuations.

Figure 1

Expanding radial growth structure and the same structure on a fixed domain with periodic boundary conditions, illustrated for the case of coalescing random walks ($\gamma =1/2$). The displacements $Y_r$ [Eq. (1)] and $X_h$ [Eq. (3)] are shown as blue (dark gray) curves. The distribution of the rescaled radial structure at radius $r$ is identical to the distribution of the fixed domain structure at height $h\left(r\right)$ as given by Eq. (8), indicated by dashed red (gray) lines. This mapping (plotted in Fig. 2) has a finite limit $h_{\gamma }\left(\infty \right)$ for $\gamma <1$, indicated by a dashed black line. Parameters are $L=100$ with $r_0=L/2\pi$, unit diffusion coefficient, and initially 100 arms.

Figure 2

The mapping $h\left(r\right)$ between an expanding radial growth structure and the same structure on a fixed domain as given in Eq. (8) in units of $r_0$ [see (11)]. Due to local equivalence of the two processes, $h\left(r\right)\approx r-r_0$ for $r\approx r_0$. The different geometries affect the behavior at large $r$, in particular, $h\left(r\right)$ has a finite limit $h_{\gamma }\left(\infty \right)$ for $\gamma <1$ (10) and diverges for $\gamma \ge 1$. The asymptotic behavior is indicated by dotted lines, except for $\gamma =2/3$, which is off the figure.

Figure 3

Numerical demonstration of the mapping (8) between radial geometry ($\circ$ on all panels) and fixed domain ($\times$): (a), (b) Number of surviving arms $\langle N\rangle$ and their root mean square distance $\langle D^2\rangle$ (15) for Lévy flights (13) with $\gamma =\max \{1/\alpha ,1/2\}$; (c), (d) the same observables for fractional Brownian motion (14); (e) number of surviving arms for Brownian motion ($\gamma =1/2$) with finite particle size $d$, $+$ symbols indicate uncorrected, $\times$ corrected particle radii (see text); (f) radial density of arms $\langle N\rangle /r$ for branching-coalescing Brownian motion with fixed branching rate $R_R$ (16) in the radial geometry. For (a)–(e) the horizontal axis is $h$ or $h\left(r\right)$, with $h\left(\infty \right)$ indicated, while for (f) the horizontal axis is $r$ or $r\left(h\right)$. The asymptotic scaling laws [black dashed lines on (a)–(d)] break down when $\langle N\rangle \approx 1$. (a)–(d) and (f) are in $1+1$ dimensions, initially 100 arms, $L=100$ with $r_0=L/2\pi$, while (e) is in $2+1$ dimensions with $r_0=20$.