About Random Growth Models

This simulation compares two classic models of random cluster growth: the Eden model and diffusion-limited aggregation (DLA). Both start with a single seed particle at the center of the grid and grow outward, but the mechanism of growth differs dramatically — producing strikingly different geometric structures.

The Eden model adds particles uniformly at random to any empty cell adjacent to the existing cluster. Because every perimeter site is equally likely to be chosen, protrusions have no advantage and the cluster remains compact, approaching a circular disk as it grows. The surface fluctuations belong to the Kardar-Parisi-Zhang (KPZ) universality class.

DLA introduces diffusion: each new particle starts far from the cluster and performs a random walk until it touches the cluster, then sticks. This creates screening — tips and branches intercept incoming walkers before they can reach fjords (inward gaps), causing the cluster to branch into a self-similar fractal with dimension D≈1.71 in 2D.

The box-counting fractal dimension and radius of gyration Rg are computed in real time and displayed in the sidebar. For DLA, D should converge toward ~1.7; for Eden, toward ~2.0 (compact disk).

Frequently Asked Questions

• What is diffusion-limited aggregation (DLA)?
  Diffusion-limited aggregation (DLA) is a cluster growth model introduced by Witten and Sander in 1981. A seed particle is fixed at the origin. Then particles are released one at a time from a random position far away and undergo a random walk (Brownian motion) until they touch the cluster, where they stick permanently. The resulting cluster is a fractal tree with characteristic branching structure and fractal dimension D≈1.71 in 2D. DLA clusters look like snowflakes, mineral dendrites, or lightning bolt paths.
• What is the Eden model?
  The Eden model (proposed by Murray Eden in 1961) grows a cluster by randomly selecting one cell from the cluster's perimeter (the set of empty cells adjacent to the cluster) and adding it. Unlike DLA, there is no long-range diffusion — growth probability is uniform over the perimeter regardless of shape. Eden clusters are compact and roughly circular with fractal surface roughness. The surface belongs to the Kardar-Parisi-Zhang (KPZ) universality class, making it relevant to surface growth in thin films, bacterial colony spreading, and tumor growth.
• Why does DLA produce fractal branching but Eden produces a compact cluster?
  The key difference is screening. In DLA, random walkers from far away are much more likely to first hit a protruding tip of the cluster than to penetrate into a fjord (an inward gap). This creates positive feedback: tips grow faster because they are more exposed to incoming walkers, while fjords are screened from walkers. This screening instability amplifies any small protrusion into a branch, then sub-branches, creating the fractal tree. Eden growth has no screening: each perimeter site has equal probability, so protrusions do not preferentially grow and the cluster stays compact.
• How is fractal dimension measured by box counting?

  Box-counting measures fractal dimension D by overlaying grids of different cell sizes ε and counting N(ε) = how many cells contain at least one cluster particle. For a fractal, N(ε) ∝ ε^(−D), so D = −lim(ε→0) log N(ε) / log ε. In practice: choose sizes ε = 1,2,4,8,...,64 pixels; count occupied boxes for each; fit a line to the log-log plot of N vs 1/ε; the slope is D. For DLA in 2D, D ≈ 1.71. For Eden clusters (solid disk), D → 2.0. For a fractal curve, 1 < D < 2.
• What is the radius of gyration of a cluster?

  The radius of gyration Rg = √(⟨r²⟩ − ⟨r⟩²) where r is the distance of each particle from the cluster's center of mass. It measures the spatial extent of the cluster. For a compact circular cluster of N particles, Rg ∝ √N (2D disk: Rg = R/√2). For a DLA fractal with dimension D, Rg ∝ N^(1/D). Measuring Rg as a function of N gives another way to estimate D: the slope of log Rg vs log N in a log-log plot equals 1/D.
• What real phenomena does DLA model?

  DLA and DLA-like growth appears widely in nature. Mineral dendrites (manganese oxide deposits that look like ferns in rock) form by DLA-like ion diffusion. Electrodeposition of zinc produces fractal zinc trees in solution. Lightning creates branched discharge channels following the path of least resistance (DLA-like). Viscous fingering (one fluid displacing another in porous rock) produces Saffman-Taylor fingers that are DLA-like in 2D geometry. Snowflake arm branching, neuron axon growth, and some tumor invasion patterns all show DLA-class fractal geometry.
• Why is the DLA fractal dimension approximately 1.71?

  The exact DLA fractal dimension in 2D has been computed numerically to D ≈ 1.710 ± 0.004, confirmed by simulations of billions of particles. An exact theoretical prediction remains one of the open problems in statistical physics. Heuristically, D is determined by the balance between diffusion (which smooths, tending toward D=2) and screening (which creates branches, reducing D below 2). In 3D, DLA dimension is D ≈ 2.50. In very high dimensions d, D → d−1 as the cluster becomes a hyper-tree.
• How is the random walk in DLA simulated efficiently?

  A naive random walk where the particle moves one step per iteration is very slow: a particle launched from radius R=100 takes ~R² ≈ 10,000 steps on average just to reach the cluster. Two optimizations are used here. First, the 'death circle': if the walker moves beyond a maximum radius (cluster_radius + safety margin), it is killed and restarted. Second, 'off-lattice jumping': when the walker is far from any cluster particle (nearest cluster particle at distance d), it can jump a random distance up to d−1 without risk of skipping over the cluster, reducing step count dramatically.
• What is the Kardar-Parisi-Zhang (KPZ) class and why is Eden in it?

  The Kardar-Parisi-Zhang (KPZ) universality class describes stochastic surface growth: the rough interface of the Eden cluster belongs to this class. The key signatures are: interface width w grows as t^β with KPZ exponent β=1/3 in 2D, and height-height correlations decay with exponent χ=1/2. This means Eden surfaces, random deposition with relaxation, bacterial colony edges, coffee stains, and even 1D TASEP particle transport all belong to the same universality class — a deep connection in non-equilibrium statistical mechanics.
• How does the number of particles affect the fractal pattern?

  In DLA, the fractal structure becomes clearer as particle count increases. Small clusters (N<100) look like irregular spidery shapes. At N≈1000 the branching hierarchy becomes visible. At N≈10,000 the self-similar structure is clear across 2–3 decades of scale. The fractal dimension D can only be reliably estimated for N>500 particles. In Eden clusters, the cluster shape approaches a circle as N increases (by the central limit theorem), but the surface roughness persists — it never becomes perfectly smooth because new fluctuations are always added at the perimeter.