Internal aggregation models on comb lattices

Abstract

The two-dimensional comb lattice C 2 is a natural spanning tree of the Euclidean lattice Z 2. We study three related cluster growth models on C 2: internal diffusion limited aggregation (IDLA), in which random walkers move on the vertices of C 2 until reaching an unoccupied site where they stop; rotor-router aggregation in which particles perform deterministic walks, and stop when reaching a site previously unoccupied; and the divisible sandpile model where at each vertex there is a pile of sand, for which, at each step, the mass exceeding 1 is distributed equally among the neighbours. We describe the shape of the divisible sandpile cluster on C 2, which is then used to give inner bounds for IDLA and rotor-router aggregation.