Eden Model vs DLA: Two Random Growth Rules, Two Shapes

Two growth rules, two very different shapes

Grow a cluster one particle at a time and the growth rule you pick decides everything about the final shape. Add particles at a uniformly random point on the cluster's boundary and you get the Eden model, proposed by biophysicist Murray Eden in 1961 as a model for how a tumour or a bacterial colony fills space: the result is compact, roughly circular, with a boundary that roughens over time but never develops holes or branches.

Let particles instead wander in from far away as random walkers and stick the instant they touch the cluster, and you get diffusion-limited aggregation (DLA), introduced by Thomas Witten and Leonard Sander in 1981. The result looks completely different: a sparse, spidery, self-similar tree of branches with empty space between them. Same basic ingredient - random particle addition - but one model gives a disc and the other gives a fractal.

Why DLA branches: the screening effect

The mechanism is a feedback loop. A random walker arriving from far away is far more likely to hit a protruding tip of the cluster than to wander all the way into a narrow gap between two existing branches - the incoming flux of walkers is highest near tips and lowest in concave regions, which is exactly the geometry of a diffusion field around an absorbing object (formally, the concentration field obeys Laplace's equation with the cluster as an absorbing boundary, so DLA is an instance of Laplacian growth). Once a tip gets slightly ahead, it captures a disproportionate share of the next walkers, grows further ahead, and shields the region behind it - a runaway instability that produces branching fractal arms rather than a smooth front.

Eden model:  pick a random site on the boundary  ->  fill it        (local rule, ignores far-away geometry)
DLA:         launch a walker, random-walk it in  ->  stick on contact (long-range diffusion field picks the site)

Measuring the fractal dimension

An Eden cluster's mass grows like its area: M ~ R^2 in two dimensions, so it has the ordinary Euclidean dimension of a disc, D = 2. Its boundary roughens according to the Kardar-Parisi-Zhang (KPZ) universality class, the same statistical family that describes burning paper fronts and bacterial colony edges - the interface width grows as a power law in time before saturating, a signature shared by a surprising range of unrelated growth processes.

A DLA cluster's mass, by contrast, grows sub-linearly with its radius: M ~ R^D with D ≈ 1.71 in two dimensions (about 2.5 in three). D is not an integer because the cluster is a fractal - self-similar branching structure at every scale, with as much empty space as filled space no matter how far you zoom in. You can estimate D directly from a simulated cluster with a box-counting or radius-of-gyration scan: plot log(mass) against log(radius) for growing sub-clusters and the slope is D.

Where DLA shows up in nature

Laplacian growth is not just a computer-science curiosity - it is the correct continuum description whenever growth is limited by diffusion of some transported quantity to a moving front. Electrodeposition experiments (metal ions diffusing to a growing electrode) produce DLA-like dendrites in the lab. Lichtenberg figures, the branching burn patterns left by high-voltage dielectric breakdown in acrylic or wood, follow the same statistics because the electric field near a conducting tip plays the role of the diffusion field. Mineral dendrites growing along rock fractures, viscous fingering when a low-viscosity fluid is injected into a high-viscosity one (the Saffman-Taylor instability), and some models of coral and bacterial-colony morphology under nutrient limitation all sit in the same universality class.

Implementing both on a grid

Both models are cheap to simulate on a lattice. The Eden model just needs a list of currently-empty boundary sites (neighbours of the cluster); pick one uniformly and fill it, then update the boundary list. DLA needs a random walker released from a large circle around the cluster, stepping until it either touches an occupied cell (stick) or wanders too far away (respawn) - killing the walker early with an escape radius keeps the simulation from spending forever on rare far excursions. Both algorithms are O(1) amortised work per added particle if the boundary/occupancy lookup is a hash set, which is why the canvas above can grow a cluster of several thousand particles in real time.

Frequently asked questions

Try it live

Everything above runs in your browser — open Random Growth and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.