Why does my box-counting D̂ differ from 1.71?

1.71 is a large-cluster extrapolation from simulations on infinite lattices; a single finite realization, a small box, and a handful of box sizes all introduce scatter. Averaging many independent clusters or pooling radii bins reduces variance, but D̂ here is intentionally lightweight pedagogy.

Is this the same algorithm as off-lattice DLA?

Same sticking rule—first contact with the aggregate—but the walk is on the square lattice with nearest-neighbor hops. Off-lattice variants move in continuous space; exponents are in the same universality class but prefactors and anisotropy differ at finite size.

Why stop near ~42% occupancy?

Once the cluster approaches the walls, boundary effects dominate and walkers spend most of their time in narrow corridors. Capping growth keeps the visualization responsive and closer to the “interior growth” regime where fractal scaling is usually quoted.

Diffusion-Limited Aggregation (DLA)

Random walkers stick on first contact with a seed cluster on a square lattice; branched fractal growth with literature mass dimension D ≈ 1.71 in 2D and a rough box-counting D̂.

About this model

Diffusion-limited aggregation (DLA) builds a random fractal cluster on a lattice: begin with a frozen seed particle, release unbiased random walkers from the system boundary (or, once the cluster is large, from an annulus just outside its bounding box), and freeze a walker the first time it occupies a site that touches the aggregate. This is the discrete analogue of diffusion fields hitting absorbing boundaries; tips of the growing object see enhanced flux, producing open, branched structures instead of compact Eden growth. In two dimensions numerical estimates of the mass fractal dimension cluster near D ≈ 1.71 in the literature (often contrasted with a mean-field Flory exponent 5/3). The simulator animates growth on a 121×121 square lattice, pauses automatically before the box is nearly full, and reports a crude D̂ from box counting on the occupied set—expect noise and finite-size bias until thousands of sites have accumulated.

Who it's for: Undergraduates in statistical physics, soft matter, or pattern formation who have met random walks and want a concrete fractal-growth model beyond percolation.

Key terms

DLA
Witten–Sander model
Random walk
Fractal dimension
Harmonic measure
Box counting

How it works

Off-lattice DLA is usually credited to Witten and Sander (1981); here you see the square-lattice variant: unbiased random walks from the boundary stick on first contact, building a branched fractal aggregate.

Key equations

Growth probability at tip scales with harmonic measure of diffusion; mean-field Flory-type exponent in 2D is D ≈ 5/3, while simulations give D ≈ 1.71. Box-counting on a finite cluster gives a noisy D̂.

Frequently asked questions

Why does my box-counting D̂ differ from 1.71?: 1.71 is a large-cluster extrapolation from simulations on infinite lattices; a single finite realization, a small box, and a handful of box sizes all introduce scatter. Averaging many independent clusters or pooling radii bins reduces variance, but D̂ here is intentionally lightweight pedagogy.
Is this the same algorithm as off-lattice DLA?: Same sticking rule—first contact with the aggregate—but the walk is on the square lattice with nearest-neighbor hops. Off-lattice variants move in continuous space; exponents are in the same universality class but prefactors and anisotropy differ at finite size.
Why stop near ~42% occupancy?: Once the cluster approaches the walls, boundary effects dominate and walkers spend most of their time in narrow corridors. Capping growth keeps the visualization responsive and closer to the “interior growth” regime where fractal scaling is usually quoted.

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Diffusion-Limited Aggregation (DLA)

About this model

How it works

Key equations

Frequently asked questions

Related simulators

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Diffusion-Limited Aggregation (DLA)

About this model

How it works

Key equations

Frequently asked questions