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 proxyP / (2√(πN)), comparing the discrete perimeterP 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.