Self-affine nature of dielectric-breakdown model clusters in a cylinder

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Abstract
Clusters grown with the dielectric-breakdown model (DBM) in cylinder geometry show two growth phases: a scaling regime for cluster heights smaller than the cylinder circumference and a subsequent steady state, which is translationally invariant in the main growth direction. By studying a very simple, exactly solvable growth model involving arrays of sticks, we arrive at an affine framework, suitably describing the scale invariance of its clusters in the scaling regime. Within this framework we derive new relations between various exponents, which are expected to also hold for DBM clusters. Numerical work on DBM in the scaling regime is presented, indicating that the clusters are indeed self-affine in this phase.