Is this the same as diffusion-limited aggregation (DLA)?

No. DLA sticks random walkers on first contact, producing fractal branches. Eden always adds a uniform perimeter site, yielding compact clusters with different scaling and morphology.

Why compare P to 2√(πN)?

For large N a smooth cluster resembles a disk of area proportional to N; its perimeter scales like 2√(πN). The ratio is a quick dimensionless roughness indicator on a finite lattice, not a fitted fractal dimension.

Can holes appear inside the cluster?

On an infinite lattice Eden growth from the outside does not enclose voids. In a finite box with absorbing walls, very late-stage geometry can be distorted; the simulator caps fill before the lattice is completely packed.

Eden Growth (Lattice)

Square-lattice Eden model: each step occupies a uniformly random empty von-Neumann neighbor of the cluster; compact growth with a rough interface—track perimeter P vs 2√(πN) as a roughness proxy.

About this model

The Eden model on Z² grows a cluster by repeatedly attaching a uniformly random empty site that touches the current cluster by a von Neumann step. It produces compact clusters whose interface fluctuates like a driven surface; the page colors cells by attachment time and reports a simple roughness proxy P / (2√(πN)), comparing the discrete perimeter P to the perimeter of a continuum disk with the same area N sites (≈1 when the boundary is smooth, >1 when wrinkled). Growth stops when the frontier empties or the box is nearly full.

Who it's for: Students learning random growth, first-passage percolation cousins, and discrete geometry of interfaces.

Key terms

Eden model
Random growth
Von Neumann neighborhood
Interface roughness
Perimeter
Lattice cluster

How it works

Eden growth builds a cluster on the square lattice by repeatedly occupying a uniformly chosen empty site that touches the cluster by a von Neumann step—compact growth with a fluctuating boundary; compare perimeter to a disk of the same area as a roughness proxy.

Key equations

A_{t+1} = A_t ∪ {x} where x ∉ A_t is chosen uniformly among sites with a neighbor in A_t (4-neighborhood). Effective perimeter P vs disk 2√(πN) at site count N.

Frequently asked questions

Is this the same as diffusion-limited aggregation (DLA)?: No. DLA sticks random walkers on first contact, producing fractal branches. Eden always adds a uniform perimeter site, yielding compact clusters with different scaling and morphology.
Why compare P to 2√(πN)?: For large N a smooth cluster resembles a disk of area proportional to N; its perimeter scales like 2√(πN). The ratio is a quick dimensionless roughness indicator on a finite lattice, not a fitted fractal dimension.
Can holes appear inside the cluster?: On an infinite lattice Eden growth from the outside does not enclose voids. In a finite box with absorbing walls, very late-stage geometry can be distorted; the simulator caps fill before the lattice is completely packed.

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Key equations

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