  Internal diffusion-limited aggregation is a growth model based on random walk
in Z^d. We study how the shape of the aggregate depends on the law of the
underlying walk, focusing on a family of walks in Z^2 for which the limiting
shape is a diamond. Certain of these walks -- those with a directional bias
toward the origin -- have at most logarithmic fluctuations around the limiting
shape. This contrasts with the simple random walk, where the limiting shape is
a disk and the best known bound on the fluctuations, due to Lawler, is a power
law. Our walks enjoy a uniform layering property which simplifies many of the
proofs.

Probability